[Math] When would a function be differentiable at the end point

Question

Let's say the function is defined on $[x,y]$

I just don't know what to think of here. I think that every function is not differentiable at the end points because they are points! How can the limit exist from the other side of the end point?

More importantly, when would a function be differentiable at the end point?

Can anyone provide an example and explain how he came with it?

Answer 1

For a continuous example take $$f(t) = \sqrt{t-x}+\sqrt{y-t}$$ where the one sided derivatives at the endpoints are infinite.