I think your teacher's statement is wrong, or your statement is incomplete. If f'(x) is discontinuous at x=a, then f'(x) is also discontinuous at x =-a.

derivative

Prove the definition of

Even function

The derivative of is

odd function

You don't need to use the continuity of f'(x), you just need to judge whether the function is at this point.

inducible

. That is to say, the derivative of even function is odd function, and the derivative function is.

Domain of definition

compare

primary function

It might be a little smaller.