Studying Mathematics Thread

[polymath257]

That is correct. if you do the *definition* of the derivative in terms of a limit for this function, you use the x^2 for the left limit and 2x-1 for the right limit. Since both limits in the definition are 2, f'(1)=2 and the function is differentiable there. It is even continuously differentiable (the derivative is a continuous function). But, it is not differentiable two times: the derivative of the derivative is not defined at x=1.

I must confess it wasn't intuitive to me at the start (mainly because one part is a linear function, the other is quadratic), but after examining the curve closely in Desmos at the point of interest, I can see how it works.

A more interesting example is f(x)=x^2 sin(1/x) for x!=0 and f(0)=0.

Here, the derivative exists everywhere, and f'(0)=0 (use the limit definition), but f'(x) is badly discontinuous.