A function 'continous' at a 'point'.

[dE_logics]

A function f(x) is continuous at x-c if corresponding to any positive number ε, arbitrarily assigned, there exists a positive number δ such that -

|f(c+h) - f(c)| < ε

for all values such that |h|<δ

This means that f(c+h) lies between f(c) - ε and f(c) + ε for all values of h lying between -δ and δ.

I was wondering that the continuity of a function is an actually function of these δs...if they are large they might cover values where the function is broken...so things actually depend on these δs and not c if δ is not infinitely small.

 

[tiny-tim]

??

that's not the way it works …

ε is given, not δ …

for a given ε, you're allowed to choose a δ as small as you like.

 

[dE_logics]

A function f(x) is continuous at x-c if corresponding to any positive number ε, arbitrarily assigned, there exists a positive number δ such that -

|f(c+h) - f(c)| < ε

for all values such that |h|<δ

Man...you have to agree this statement is very badly written...I can't understand a thing (now
)!

Can someone explain??

 

[dE_logics]

Yes also, suppose in the continuous function f(x) can it happen that for an infinity small change in x the value of f(x) does not change?

 

[emyt]

if you want to examine a limit as x->a, f(a) is the limit, so you want to pick an interval of points around that limit.. say, an interval of 1. and then there will exist corresponding x values for that interval. if you picked the delta first, who knows what you'll get?

if you ever think about how "for every epsilon there exists a delta" works, think about a function every y there exists x such that ƒ(x) = y

 

[statdad]

The idea behind saying that

$$|f(c+h) - f(c)| < \epsilon \qquad \text{ for } |h| \le \delta$$

is that when you are "really close" to c, function values are "really close" to f(c).

In general (not always) the smaller the value of $$\epsilon$$ you select, the smaller must be $$\delta$$

Think about this geometric approach. Draw a portion of an arbitrary continuous function (draw any continuous curve) - for a specific example, draw it near $$c =2$$, and suppose $$f(2) = 5$$.

Now pick $$\epsilon = 0.05$$ and draw the two horizontal lines $$y = 5 - \epsilon$$ and $$y = 5 + \epsilon$$.
Now draw two vertical lines, one on each side of $$c = 2$$ (equally spaced) so that
all of the graph between these two lines is between the two horizontal lines you drew at the first step. The common distance these vertical lines are from 2 is $$\delta$$ - so, given $$\epsilon = .05$$, you've just found a $$\delta > 0$$ such that

$$|f(2+h) - f(2)| < \epsilon \qquad \text{ for } |h| < \delta$$

Note: I've just made an example with Latex and a few lines of tikz code. It's attached.

 

[dE_logics]

if you want to examine a limit as x->a, f(a) is the limit, so you want to pick an interval of points around that limit.. say, an interval of 1. and then there will exist corresponding x values for that interval. if you picked the delta first, who knows what you'll get?

if you ever think about how "for every epsilon there exists a delta" works, think about a function every y there exists x such that ƒ(x) = y

I absolutely do not know limits...but I did understand what you said (sorta).

I find the variable δ very reluctant...why not skimpily manipulate the value of h?...why make the relation h = δ and THEN manipulate h?

So over all this |f(c+h) - f(c)| < ε && |h|<δ mechanism is trying to say that for an infinitely small change in value of h in the function f(x+h); the value of f(x+h) will also change by an infinity small value within the interval [a, b] (corresponding to value x+h) if the function is called continuous within [a, b]...it's a test for continuity.

By this we can also define the continuity at the 'point' f(x+h) cause here the value of h is infinitely small...so WHY h, δ, x, ε...SO many variables?

Anyway...I do not even understand the definition of the variables ε...it's neither f(x) nor f(x-h)...what is it (theoretically)?

 

[lurflurf]

^
wow!
Using δ instead of h has three purposes
1)Greek letters are really cool and fun to write.
2)δ>|h| so δ is a ristriction on h not its value
3)δ establishes a short hand that, one knows what it means in different situations.

Infinitely small is a strange term, we have a better one infinitesimal.
h, δ, ε are note infinitesimal they are small enough.
In words the definition of a limit L=lim_x->a f(x) would be
"The function f can be made as close to L as desired by chosing x sufficiently close to a"
or for f is continuous at a
"The function f can be made as close to f(a) as desired by chosing x sufficiently close to a"

ε the maximum allowed difference between the function and the limit

Think of it like a traffic law. Your driving around in you facy function f and you see a sign "|f(x+h)-f(x)|<ε". If f is contimuous you can manage this by making sure |h|<δ so you will not get a ticket.

 

[dE_logics]

Latex is not working on your signature.

I'm inept of spelling 'infinitesimal'...my tongue sort of grows a knot in it if I try to speak it; so I prefer listening/reading it rather than spelling/typing it.

Man...I'm not getting a thing...problem is I do not know the definitions...like what is h; for starters...I know it's not an arbitrary which, to me, seems very likely...so can someone please tell.

 

[dE_logics]

Formally, f (x) is defined to be continuous if for any real x and any infinitesimal dx, f (x + dx) − f (x) is infinitesimal.

So my guess was right...that complex 4 letter definition meant the same thing...thanks to recoll, I got this ebook.

 

[dE_logics]

But finally I realized that the ε and δ are pretty much standard...so I should know about them.

So what is h?

 

[Office_Shredder]

h is just a number smaller than delta.

What the definition of continuity is saying is:

If I give you some small value epsilon, you can give me a small value of delta so that if y is within delta of x, f(y) is within epsilon of f(x). It's like a game... your goal is to show something is continuous by always giving me a delta back when I give you epsilon

 

[dE_logics]

And what is delta?

 

[tiny-tim]

ε is a circle (or "neighbourhood") in the image space, δ is a circle (or "neighbourhood") in the object space, and h is anything inside δ.

So, for any circle in the image space, you have to find a circle in the object space whose image lies completely inside it. ​

 

[dE_logics]

Image and object space? ..I don't get a thing.

Ok, this is what I pondered out -

First we will like to have the list of variables -
ε – An 'arbitrary' value which is supposed to be related to f(x)...in some way...after including a variable 'h'.

x – X server...just kidding. The independent variable of the function f.

h – A value added/subtracted to x; since it will be added/subtracted to x, f(x + h) or f(x – h), it will return a deviated value as compared to f(x). It's a criteria that the deviation should be < ε...in mathematical terms |f(x+h)-f(x)| < ε or |f(x-h)-f(x)| < ε. So actually this value (h) has to be computed keeping the value of ε in mind.

A function is said to be continuous iff for a very small value of ε the corresponding value of h is also small; notice that by this definition, even if |f(x+h)-f(x)| = 0, the function will be considered continuous.
If taking a fresh example of the function g(e) = y; for an infinitely small change in value of e, there should be an infinitely small change or no change in value of y for the function to be continuous.

Most probably this is wrong cause I found the need of another variable, δ unnecessary.

 

[Born2bwire]

h is not computed with \epsilon in mind. h is arbitrary, what we are specifying is that there is a continuous range from -h to h inclusive by which the distance of f(x+h) from f(x) is always less than \epsilon. We are interested in the behavior of the function locally about x, we do not care what h is, we only care about the restrictions on h and the restrictions on the change in the function when we deviate by some h.

 

[dE_logics]

h is not computed with \epsilon in mind. h is arbitrary, what we are specifying is that there is a continuous range from -h to h inclusive by which the distance of f(x+h) from f(x) is always less than \epsilon. We are interested in the behavior of the function locally about x, we do not care what h is, we only care about the restrictions on h and the restrictions on the change in the function when we deviate by some h.

Thanks A LOT man...that helped my by A LOT...finally something clear that I can understand.

So we can say that |f(x+h) - f(x)| and |f(x-h) - f(x)| should not exceed ε and ε is the constant here. Since |f(x+h) - f(x)| and |f(x-h) - f(x)| is also a function of h, ε poses a restriction on h.

But this is sort of defining the limit...not continuity; what do we say for continuity?

 

[Office_Shredder]

Thanks A LOT man...that helped my by A LOT...finally something clear that I can understand.

So we can say that |f(x+h) - f(x)| and |f(x-h) - f(x)| should not exceed ε and ε is the constant here. Since |f(x+h) - f(x)| and |f(x-h) - f(x)| is also a function of h, ε poses a restriction on h.

But this is sort of defining the limit...not continuity; what do we say for continuity?

An equivalent definition of continuity at a point is f(x) is continuous at c if$$\lim_{x \rightarrow c} f(x) = f(c)$$

Can you see how this matches up with the epilon delta definition?

 

[dE_logics]

No, cause I do not understand the notion of limit.

Can you please tell me in terms of ε and δ?

 

[dE_logics]

 

[HallsofIvy]

You keep saying you "do not know limits". How can you hope to understand, or even ask about, continuity, then?

But since you ask: "$\lim_{x\to a} f(x)= L$" if and only if, given any $\epsilon> 0$ there exist $\delta> 0$ such that if $0< |x- a|< \delta$, then $|f(x)- L|< \epsilon$.

That basically means, as just about every other response here has said, that you can f(x) as close to L as you like just by taking x close enough to a.

In order that a function, f(x), be continuous at x= a, three things must be true:
1) f(a) must exist.
2) $\lim_{x\to a} f(x)$ must exist.
3) $\lim_{x\to a} f(x)$ must be equal to f(a).

Since just writing $\lim_{x\to a} f(x)= f(a)$ pretty much implies that the two sides exist, most of the time we just write that.

 

[statdad]

You stated that $$f$$ is continuous at $$c$$ if, for every $$\epsilon > 0$$, there is a $$\delta > 0$$ such that

$$| f(c+h) - f(c)| < \epsilon \quad \text{ if } |h| < \delta$$

An intuitive statement of continuity at a point is this:

The function $$f$$ is continuous at a number $$c$$ if $$f(c+h)$$ is
close in value to $$f(c)$$ when $$c+h$$ is close to $$c$$.

• $$\epsilon$$ shows how close you want $$f(c+h)$$ to be to $$f(c)$$
• $$\delta$$ measures how close to $$c$$ you need to pick other x-values to make the previous point come true
• $$h$$ measures the distance the new x-value is from the number $$c$$

 

[dE_logics]

You keep saying you "do not know limits". How can you hope to understand, or even ask about, continuity, then?

That's cause in every book first they explain continuity, then limit.

You stated that $$f$$ is continuous at $$c$$ if, for every $$\epsilon > 0$$, there is a $$\delta > 0$$ such that

$$| f(c+h) - f(c)| < \epsilon \quad \text{ if } |h| < \delta$$

An intuitive statement of continuity at a point is this:

The function $$f$$ is continuous at a number $$c$$ if $$f(c+h)$$ is
close in value to $$f(c)$$ when $$c+h$$ is close to $$c$$.

• $$\epsilon$$ shows how close you want $$f(c+h)$$ to be to $$f(c)$$
• $$\delta$$ measures how close to $$c$$ you need to pick other x-values to make the previous point come true
• $$h$$ measures the distance the new x-value is from the number $$c$$

PERFECT answer man...just perfect...thanks.

 

[HallsofIvy]

That's cause in every book first they explain continuity, then limit.

In every book? Peculiar- I have never seen such a book!



PERFECT answer man...just perfect...thanks.[/QUOTE]
Yes, it was.

 

[dE_logics]

Yes, check out these books -

DIFFERENTIAL CALCULUS - SANTI NARAYAN
Introduction to calculus - KAZIMIERZ KURATOWSKI
Calculus - Benjamin Crowell

Actually there are no books where limits is discussed before continuity.

So the value of h has 2 restrictions...one ε and the other δ.

Is the value of δ arbitrator?

 

[Office_Shredder]

The value of delta depends on the value of epsilon. The value of epsilon is arbitrary.

I'm looking at Introduction to calculus by Kuratowski right now. Section 4: Functions and their Limits

Section 5: Continuous functions

I notice that it defines limits in terms of sequences but these are equivalent. At any rate, the book definitely defines limits before continuity

 

[dE_logics]

hummm...yes, you're right about Kuratowski.

Exclude that book and Benjamin Crowell has continuity at 2.8 and limits at 2.9

 

[emyt]

Yes, check out these books -

DIFFERENTIAL CALCULUS - SANTI NARAYAN
Introduction to calculus - KAZIMIERZ KURATOWSKI
Calculus - Benjamin Crowell

Actually there are no books where limits is discussed before continuity.

So the value of h has 2 restrictions...one ε and the other δ.

Is the value of δ arbitrator?

Calculus - Michael Spivak
Introduction to Calculus and Real Analysis - Richard Courant and John Fritz

both these books are popular, and they have a treatment on limits before continuity

 

[dE_logics]

I do have Introduction to Calculus and Real Analysis - Richard Courant and John Fritz somewhere...anyway, I'll get them.

Thanks.

 

[dE_logics]

This is the verdict; hope it helps someone -

Consider an example...lets have a function such that for an interval a to b the function returns values continuously but below a or below b it returns one values (for the respective limit cross (below a and above b)).
First we will like to have the list of variables -
ε – An 'arbitrary' value which is supposed to be related to f(x)...in some way...after including a variable 'h'.
x – X server...just kidding. The independent variable of the function f.
δ – 'h' is made to assume this value actually such that |h|<δ; δ is also dependent on ε and it's value is specific. The value of h can be varied arbitrarily, but it has a restriction |h|<δ...this is the limit of h's arbitrary value.
h – A value added/subtracted to x; since it will be added/subtracted to x, f(x + |h|) or f(x – |h|), it will return a deviated value as compared to f(x). It's a criteria that the deviation should be < ε...in mathematical terms |f(x+|h|)-f(x)| < ε or |f(x-|h|)-f(x)| < ε; notice that|h|<δ, the value of δ is such that it will not permit |f(x+h)-f(x)| or |f(x-|h|)-f(x)| to be greater and equal to ε...or the value of |h| should not exceed δ. So the value of δ is based on the restriction posed by ε and |h| can be varied by |h|<δ.
A function is said to be continuous iff for a very small value of ε the corresponding value of δ is also small and cause value of δ will be small the rigidity in value of h will too be high; notice that by this definition, even if |f(x+|h|)-f(x)| = 0, the function will be considered continuous.
If taking a fresh example of the function g(e) = y; for an infinitely small change in value of e, there should be an infinitely small change or no change in value of y for the function to be continuous.

 

[dE_logics]

aaaa...but still there is a major doubt left.

What do you mean by a 'point'...for a function to be continuous at a 'point', the value of ε should be assumed infinitely small and the corresponding value of δ should also be infinitely small...THEN we can call the function continuous at a 'point'...and that point will be f(x)...or the value of f at x.

 

[emyt]

aaaa...but still there is a major doubt left.

What do you mean by a 'point'...for a function to be continuous at a 'point', the value of ε should be assumed infinitely small and the corresponding value of δ should also be infinitely small...THEN we can call the function continuous at a 'point'...and that point will be f(x)...or the value of f at x.

if a function is continuous at a point, then there will exist some kind of neighbourhood around that point such that f(a+h) as h approaches 0 is f(a).

-------xxxxxxx--f(a)--f(a+h)--xxxxxx------xxxxx -------

imagine this line as a discontinuous function (the x's are spaces, spaces don't work here), the dashes can be infinitesimally small distances, but the function is continuous at a because there is a neighbourhood around it where you can pick f(a+h) and have the limit of that as h goes to zero as f(a) ----- xxxxx-- xxx-------xxxxxxxf(a)xxxxxxx---- xxxxx--------- -----xxxxx ------


if f(a) is defined like this, then there is no neighbourhood around f(a) such that f(a+h) as h approaches 0 is f(a)----- --- ------- xxxxxxf(a)xxxxxxxx----f(a+h)-xxxxxxxx------------ ------ ------
----- --- ------- xxxxxxxx f(a)xxxxxxxx--f(a+h closer..)---xxxxxxxx ------------ ------ ------ ----- --- -------xxxxxxxxxx f(a)xxxxxxxxxx-f(a+h cannot go any closer)---- ------------ ------ ------

 

[HallsofIvy]

aaaa...but still there is a major doubt left.

What do you mean by a 'point'...for a function to be continuous at a 'point', the value of ε should be assumed infinitely small and the corresponding value of δ should also be infinitely small...THEN we can call the function continuous at a 'point'...and that point will be f(x)...or the value of f at x.

There is no such thing as "infinitely small" real numbers. You can do calculus in terms of "infinitesmals" but that requires extending the real numbers to a new number system and that is very deep mathematics. Certainly nothing you have said so far implies that you are familiar with infinitesmals and I recommend avoiding them in favor of the "limit" concept we have been using so far.

Saying that a function is "continuous at a point", say "f(x) is continuous at x= a", is exactly what we have been talking about here. "f(x) is continuous at x= a" if and only if
1) f(a) exists.
2) $\lim_{x\to a}f(x)$ exists.
3) $\lim_{x\to a} f(x)= f(a)$.

More fundamentally, including the definition of "limit" in that definition
"Given any $\epsilon> 0$, there exist $\delta> 0$ such that if $|x- a|< \delta$ then $|f(x)- f(a)|< \epsilon$".

The usual definition of "continuous" is "continuous at a point". We then extend the concept by saying that f(x) is "continous on a set" if and only if it is continuous at every point of that set.

Saying that a function is continuous "at a point" does not restrict the possible values of $\delta$ and $\epsilon$ in any way.