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We learned last class, if I remember correctly, that a function cannot be differentiable at endpoints of an interval because the two-sided limit of the difference quotient can't exist. Now we are told the function $g$ is differentiable on the same closed interval on which it is defined. What gives?

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Is what we learned last class incorrect, is the above problem incorrect, or is my interpretation of either incorrect? Any ideas?

Differentiability and continuity at endpoints has always been a topic that confuses me, even though I think I understand the concepts at interior points. I seem to remember us learning last class that differentiability requires being able to approach a point from both sides, which you can't do at endpoints, right?

Will
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When you want to check if a function is differentiable at its endpoints you only need to check if the $one-sided$ limit exists and is a real number. That's because the two sided limit isn't even defined for the endpoints.

When you want to check if a function is differentiable at an interior point, you need to check the $two-sided$ limit (which needs to exist and be a real number in order for your function to be differentiable at this point).

This answer will help you too.

I found the link after writing the first two mini-paragraphs of my answer and I think that this is the most common definition given (and the one given to me at school by my teachers), so I will leave them(the two paragraphs) here.

michalis vazaios
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