Does this sketch meat the given requirements?

Integrate

Integrate

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Draw the graph of a function on [0, 5] with the given properties:

f (x) is not continuous at x = 1, but [math]\lim_{x \to 1^+} f(x)[/math] f (x) and [math]\lim_{x \to 1^-} f(x)[/math] exist and are equal.




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From what I understand the definition of a limit existing is as follows

[math]\lim_{x \to a^-} f(x) = f(a) = \lim_{x \to a^+} f(x)[/math]
meaning that the both sides of the limit must approach f(a) but do not necessarily need to equate to each other.

Allowing us to never define what happens at f(a).

In other words I can just leave f(a) blank if I so desire.

I have made a very crude sketch.

IMG_1586.jpg

Does this fit the given requirements?


I ask this question because the textbook solution does in fact give f(a) a value.
 
Integrate said:
From what I understand the definition of a limit existing is as follows

[math]\lim_{x \to a^-} f(x) = f(a) = \lim_{x \to a^+} f(x)[/math]
meaning that the both sides of the limit must approach f(a) but do not necessarily need to equate to each other.
Click to expand...
What you wrote symbolically (which I don't think is what you meant) is the definition of a continuous function, not of a limit existing. The limit doesn't care what the value of f(a) is, or whether it exists; continuity does.

So for a limit, there is no need to approach f(a), but there is a need to approach the same value from both sides. What you wrote in words is wrong for both limit and continuity. But you're right in what follows ...

Integrate said:
Draw the graph of a function on [0, 5] with the given properties:

f (x) is not continuous at x = 1, but [math]\lim_{x \to 1^+} f(x)[/math] f (x) and [math]\lim_{x \to 1^-} f(x)[/math] exist and are equal.

...

Allowing us to never define what happens at f(a).

In other words I can just leave f(a) blank if I so desire.

I have made a very crude sketch.

View attachment 31280

Does this fit the given requirements?

I ask this question because the textbook solution does in fact give f(a) a value.
Click to expand...
I think the reason they do assign a value to f(a) is that they say "a function on [0, 5]", which means it must be defined on the entire interval. Otherwise, you would be right (if the hole were at x=1).
 
Will refine my vocabulary. You are right that is the definition of continuity I wrote out. Thank you!

What are the requirements that a limit needs to exists, again? Google search is coming up noisy.
 
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