# Showing that a function is continuous

## Homework Statement

How would I demonstrate that a function is continuous? Would I just show that it's derivative exists? Thanks for the help.

Mark44
Mentor

## Homework Statement

How would I demonstrate that a function is continuous? Would I just show that it's derivative exists? Thanks for the help.
But what if the function is not differentiable? There are functions that are not differentiable at a given point that happen to be continuous there. For example, y = |x| at x = 0.

Then it would be where it's undefined? Also, would I have to examine its end behavior?

Mark44
Mentor
No. My example function, f(x) = |x|, is defined everywhere.

How is continuity defined in your book?

It's defined by saying that a function is continuous if its graph is an unbroken curve over an interval

Mark44
Mentor
The way continuity at a point is usually defined is as a limit:
A function f is continuous at a number a in its domain iff
$$\lim_{x \to a}f(x) = f(a)$$