Continuous Functions and Differentiability

Here is an interesting question about derivatives posted by an anonymous reader. Is every continuous function differentiable? Is every differentiable function continuous?

The answer to the first question is in the negative. A simple example of a function that is continous but not differentiable is the function defined on the reals:

Details of why this function is continous but not differentiable can be found here: functions with corners.

The answer to the second question is positive. The proof of this is very simple. According to our definition of continuity, a function is continous when:

Or equivalently:

If f is differentiable, we have that:

And that completes the proof.

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