Understanding Right-Continuity in Piecewise Functions: What is g(c)?

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In summary: A function is left-continuous if it stays the same when you go one direction only (like going left on the y-axis), while a function is right-continuous if it stays the same when you go both directions (like going up and down the y-axis).In summary, the discontinuity at x=1 is a jump discontinuity, but it is left-continuous. The discontinuity at x=5 is a jump discontinuity, but it is right-continuous.
  • #1
phrox
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I don't really understand this question...

I'm given a graph with x approaching and hitting 1, making y=2 (filled dot). Then there's a discontinuity jump at (1,3) which is the empty dot, then there's 2 other points on this small curve with empty dots at (3,4.5) and at (5,4), then another jump to a filled dot at (5,1).

For the first question being find point c that the function has jump discontinuity but is left-continuous which I know would be c=5.

It's the second question that doesn't make sense to me. It is:
What value should you assign to g(c) to make f right - continuous at x=c?

This question just doesn't make sense to me. What is g(c) on the graph? The empty dotted curve? Would by "f" he meant the whole f(x) function which has discontinuity?

Thanks!
 
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  • #2
Can you include an image of the graph?
 
  • #3
Sure thing,

http://i1301.photobucket.com/albums/ag115/phrox1/graph_zps49a57ee8.png
 
  • #4
phrox said:
http://i1301.photobucket.com/albums/ag115/phrox1/graph_zps49a57ee8.png

phrox said:
Then there's a discontinuity jump at (1,3) which is the empty dot, then there's 2 other points on this small curve with empty dots at (3,4.5) and at (5,4), then another jump to a filled dot at (5,1).
For the first question being find point c that the function has jump discontinuity but is left-continuous which I know would be c=5.
It's the second question that doesn't make sense to me. It is:
What value should you assign to g(c) to make f right - continuous at x=c?
This question just doesn't make sense to me. What is g(c) on the graph? The empty dotted curve? Would by "f" he meant the whole f(x) function which has discontinuity?

I do not really follow some of this question, but the graph helps.
The function \(\displaystyle f\) has a jump discontinuity at \(\displaystyle x=1\) but it is left-continuous there.
The function \(\displaystyle f\) has a jump discontinuity at \(\displaystyle x=5\) but it is lright-continuous there.
See this web page.

The function \(\displaystyle f\) has an discontinuity at \(\displaystyle x=3\) that some call a removable discontinuity .
 
  • #5
If g(x) = f(x) except for at points of discontinuity of f, then to make g right-continuous at c = 1 (which is the only place where f has a jump discontinuity but is left-continuous, your answer of c = 5 is wrong) we must choose g(1) = 3.

This is just a conjecture on my part, you haven't told us what "g" is. A summary of the discontinuities at f:

x = 1:

f is left-continuous, and has a jump discontinuity

x = 3

f is neither left-continuous nor right-continuous and has a jump discontinuity. if we were to replace the value f(3) = 2.5 (approx.) with f(3) = 4.5 (approx., it is REALLY hard to tell from your graph) f would be continuous at 3 (which is why the continuity is called "removable")

x = 5

f is right-continuous and has a jump discontinuity.

Neither the discontinuity at 1, nor at 5 can be fixed by assigning some "other" value for f(x), the resulting function will still be discontinuous there (sometimes these are called "essential" discontinuities).
 
  • #6
I do agree with that. But for the second question, it doesn't make sense because there has to be an answer according to my prof. He said if it doesn't exist, put DNE. But that doesn't work.

Also, 1) is correct at c=5 because it just simply says it's correct when I have it right.
 
  • #7
It? What is this "it"?

If by "it" you mean some automated software on a computer-graded test/problem set, I can assure you "it" most certainly can be wrong:

1) The program could have an internal error.
2) Whichever human supplied the "answer data file" could have made a mistake.

In general, mathematical statements are not true because someone or something says they are, they are true because they are logically consistent with, and dependent upon other facts or axioms given before-hand. "Proof by authority" is NOT accepted by the mathematical community at large.

IF (and this is actually a big "if") you have portrayed the graph of f(x) reasonably accurately (in particular, if the holes are solid or "hollow" at the "special points" as you depict it) then f is NOT left-continuous at c = 5.

I state the formal definition of left-continuous here:

A function $f:\Bbb R \to \Bbb R$ is said to be left-continuous at $a$ if:

$\displaystyle \lim_{x \to a-} f(x) = f(a)$.

From the graph you have supplied, we see that:

$\displaystyle \lim_{x \to 5-} f(x) = 4$

but $f(5) = 1$

As 4 does not equal 1, the matter is settled, and anyone who says differently is just wrong.

Perhaps we are missing some crucial information here, but it sounds like someone has confused left with right.
 

FAQ: Understanding Right-Continuity in Piecewise Functions: What is g(c)?

What is meant by continuity for piecewise functions?

Continuity for piecewise functions refers to the property of a function being continuous at every point within its domain, even if the function is defined by multiple pieces or segments. This means that there are no sudden breaks or jumps in the graph of the function.

How do you determine if a piecewise function is continuous?

To determine if a piecewise function is continuous, we must check for continuity at each point where the pieces or segments meet. This involves evaluating the limit of the function at that point and comparing it to the value of the function at that point. If the limit and the function value are equal, then the function is continuous at that point.

What happens if a piecewise function is not continuous at a point?

If a piecewise function is not continuous at a point, it means that there is a break or jump in the graph at that point. This can happen if the limit and the function value at that point are not equal. In this case, the function is said to have a point of discontinuity at that point.

Can a piecewise function be continuous at some points and discontinuous at others?

Yes, it is possible for a piecewise function to be continuous at some points and discontinuous at others. This depends on the specific pieces or segments of the function and how they connect at the points of intersection. It is important to check for continuity at each point where the pieces meet to determine the overall continuity of the function.

What is the difference between continuity and differentiability for piecewise functions?

Continuity and differentiability are related but distinct concepts for piecewise functions. Continuity refers to the smoothness of the graph of the function, whereas differentiability refers to the existence of a derivative at a point. A function can be continuous at a point without being differentiable, and vice versa.

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