In mathematics, a continuous function is a function that does not have any abrupt changes in value, known as discontinuities. More precisely, a function is continuous if arbitrarily small changes in its output can be assured by restricting to sufficiently small changes in its input. If not continuous, a function is said to be discontinuous. Up until the 19th century, mathematicians largely relied on intuitive notions of continuity, during which attempts such as the epsilon–delta definition were made to formalize it.
Continuity of functions is one of the core concepts of topology, which is treated in full generality below. The introductory portion of this article focuses on the special case where the inputs and outputs of functions are real numbers. A stronger form of continuity is uniform continuity. In addition, this article discusses the definition for the more general case of functions between two metric spaces. In order theory, especially in domain theory, one considers a notion of continuity known as Scott continuity. Other forms of continuity do exist but they are not discussed in this article.
As an example, the function H(t) denoting the height of a growing flower at time t would be considered continuous. In contrast, the function M(t) denoting the amount of money in a bank account at time t would be considered discontinuous, since it "jumps" at each point in time when money is deposited or withdrawn.
I was just reflecting upon my math courses and wondered why can we transform any piecewise continuous functions by using transforms such as laplace transforms or converting to Fourier series by simply adding the required integrals on the respective bounds?
Ok, so this was assigned as a bonus problem in my Topology class a while ago. Nobody in the class got it, but I've still been racking my brain on it ever since.
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For some n, consider the set of all nxn nonsingular matrices, and using the usual Euclidean topology on this space, show that...
Let f be a uniformly continuous function on Q... Prove that there is a continuous function
g on R extending f (that is, g(x) = f(x), for all x∈Q
I think I am supposed to somehow use the denseness of Q and the continuity of a function to prove this, but I am not quite sure where I should start...
Homework Statement
If f and g are continuous functions, with f(3) = 5 and \stackrel{lim}{x\rightarrow3}\left[2f(x) - g(x)\right] = 4 find g(3)
The Attempt at a Solution
I'm stumped! I cannot find anything in my notes on where to begin. I am not looking for a specific answer, I just need...
Homework Statement
Let F be the set of all continuous functions with domain [-1,1] and codomain R. Let A be the algebra of all polynomials that contain only terms of even degree (A is a subset of F). Show that the closure of A in F is the set of even functions in F.
The attempt at a...
Homework Statement
Let f and g be continuous functions defined on all of R. Prove that if f(a) \neq g(x) for some a \epsilon R , then there is a number \delta > 0 such that f(x) \neq g(x) whenever |x-a| < \delta.
Homework Equations
I would like to please check if my proof is...
Let I be the interval I=[0,infinity). Let f: I to R be uniformly continuous. Show there exist positive constants A and B such that |f(x)|<=Ax+B for all x that belongs to I.
Please help me!~
Homework Statement
assume h: R->R is continuous on R and let K={x: h(x)=0}. Show that K is a closed set.
Homework Equations
The Attempt at a Solution
since we know h is continuous and h(x)=0. therefore, we know there is a epsilon neighborhood such that x belongs to preimage...
Homework Statement
Let Ce([0,1], R) be the set of even functions in C([0,1], R), show that Ce is closed and not dense in C.
Homework Equations
The Attempt at a Solution
I think I can solve this if I can show that even functions converge to even functions, but I can't quite...
Hi,
I think I've gotten this problem, but I was wondering if somebody could check my work. I still question myself. Maybe I shouldn't, but it feels better to get a nod from others.
Homework Statement
Consider the space of functions C[0,1] with distance defined as...
Homework Statement
A mapping f from a metric space X to another metric space Y is continuous if and only if f^{-1}(V) is closed (open) for every closed (open) V in Y.
Use this and the metric space (X,d), where X=C[0,1] (continuous functions on the interval [0,1]) with the metric d(f,g)=\sup...
This is a question from Papa Rudin Chapter 2:
Find continuous functions f_{n} : [0,1] -> [0,\infty) such that f_{n} (x) -> 0 for all x \in [0 ,1] as $n -> \infty. \int^{1}_{0} f_n dx -> 0 , but \int^{1}_{0} sup f_{n} dx = \infty.
Any idea? :) Thank you so much!
Let X be a compact space, (Y,p) a compact metric space, let F be a closed subset of C(X,Y) (the continuous functions space) (i guess it obviously means in the open-compact topology, although it's not mentioned there) which satisifes:
for every e>0 and every x in X there exists a neighbourhood U...
Homework Statement
How is the theorem "Continuous functions have closed graphs" proven in the setting of a general topological space? (assuming the theorem is still valid?)
I'm having trouble with the third part of a three part problem (part of the problem is that I don't even see how what I'm trying to prove can be true).
The problem is:
Let X and Y be topological spaces with X=E u F. We have two functions: f: from E to Y, and g: from F to Y, with f=g on the...
Homework Statement
Okay, so if f and g are continuous functions at a, then prove that f/g is continuous at a if and only if g(a) # 0
Homework Equations
Assuming to start off the g(a)#0, by the delta-epsilon definition of continuity, basically, We know that |f(x)| and |g(x)| are bounded...
Does the ring of continuous functions over the real numbers have no zero divisors? If no 0 divisor, how can I prove it? Else, what is a counter example?
Homework Statement
In topology, a f: X -> Y is continuous when
U is open in Y implies that f^{-1}(U) is open in X
Doesn't that mean that a continuous function must be surjective i.e. it must span all of Y since every point y in Y is in an open set and that open set must have a pre-image...
Homework Statement
We recently proved that if a function, f, is continuous, it's absolute value |f| is also continuous. I know, intuitively, that the reverse is not true, but I'm unable to come up with an example showing that, |f| is continuous, b f is not. Any examples or suggestions would...
This is not a homework but it is a question in my mind.please guide me.
Let X and Y be topological spaces,let f : X -----> Y is a function.
when the following statements are equivalent?:
1) f is continuous
2) f(A') is subset of f(A)' ,for every A subset of X.
Symbols: A' i.e...
i need to find the cardinality of set of continuous functions f:R->R.
well i know that this cardinality is samaller or equal than 2^c, where c is the continuum cardinal.
but to show that it's bigger or equals i find a bit nontrivial.
i mean if R^R is the set of all functions f:R->R, i need to...
After reading http://en.wikipedia.org/wiki/Weierstrass_function it occurred to me that I could do the same thing to an integral:
\int \sum_{i=0}^\infty \frac{sin(\frac{x}{3^i})}{2^i} dx
= \sum_{i=0}^\infty \int \frac{sin(\frac{x}{3^i})}{2^i} dx
= \sum_{i=0}^\infty...
This is a simple question: given a continuous function, f(x), in a closed interval, how do we show that there is value "a" small enough such as for arbitrary x:
f(x+a) - f(x) < e
Where e lower bound is 0.
?
Thxs in advance.
Let f: D \rightarrow \mathbb{R} be continuous.
Is there an easier function that counterexamples;
if D is closed, then f(D) is closed
than D={2n pi + 1/n: n in N}, f(x)=sin(x) ?
Plus, these counterexamples are all the same with the domain changed, just correct me if I'm wrong.
If...
The question reads: Is it true that every compact subset of \mathbb{R} is the support of a continuous function? If not, can you describe the class of all compact sets in \mathbb{R} which are supports of continuous functions? Is your description valid in other topological spaces?
The answer to...
"Let h and g be uniformly continuous on I\subset\mathbb{R} and are both bounded. Show that hg is uniformly continuous."
h and g are uniformly continuous and bounded on I implies that h and g are continuous and bounded on I. This implies that hg (h times g) is continuous and bounded on I. Let...
hi,
My question reads:
Let f be defined and continuous on the interval D_1 = (0, 1),
and g be defined and continuous on the interval D_2 = (1, 2).
Define F(x) on the set D=D_1 \cup D_2 =(0, 2) \backslash \{1\} by the formula:
F(x)=f(x), x\in (0, 1)
F(x)=g(x), x\in (1, 2)...
I need to find a value for f at (0,0) to make this function continuous:
f(x,y)=sqrt(x^2+y^2)/[abs(x) + abs(y)^(1/3)]
With other functions in this problem I simply took the limit .. but taking the limit gives 0/0. In single-variable calculus I would apply l'hopital's rule to this, but I'm...
I would like a proof or a counter-example for the following claim:
A non-constant real-valued continuous function (f:R->R) cannot have an arbitirarly small period!
There are several definitions of a continuous function between metric spaces. Let (X,d_X) and (Y,d_Y) be metric spaces and let f:X\rightarrow Y be a function. Then we have the following as definitions for continuity of f:
\square \quad \forall\, x \in X \mbox{ and } \forall \, \epsilon >0 \...
Let D[a,b] be the set of piece-wise continuous functions on the close interval [a,b]. Show that D[a,b] is a subspace of the vector space P[a,b] of all functions defined on the interval [a,b].
Can someone get me started? Do I just need to show that they are closed under addition and...
Cb is set of all complex valued bounded functions
R is set of Real numbers
Define F:Cb(R)->Cb(R) by F(f)=f^2 for all 'points' f is an element of Cb(R). Prove that F is continuous.
Can someone give me some guidance on how to get started with this one?
lo,
I've got a quick q about the equation in the title, I've been asked to show/prove by analysis, that if f and g are continuous functions then M(x) is also continuous, it seems pretty intuitive but i just don't know how they want us to prove it, any help would be gr8
I want to determine all continuous functions s.t. for all x, y reals:
f(x+y)f(x-y) = {f(x)f(y)}^2
Now, I want to know where on the web I can learn how I go about doing this, because I don't know what methods to use or what I should be aiming at in manipulating the above.
I don't want...