Originally Posted by

**aman_cc** In many textbooks I have read a statement like this -

if *f'(x) exists at x=x0* **and** *is continuous at x=x0* then => some follow-up logic

My question is

Doesn't the existence of f'(x) (first derivative) at x=x0 imply it is continuous at x=x0? I say that because of the way f'(x) is defined at x=x0.

Also if f'(x) is continuous at x=x0 then it obviously it exists at x=x0.

Hence the two statements: 1. *f'(x) exists at x=x0 *2. f'(x) is continuous at x=x0 are **equivalent**

Am I correct? Or I am missing something?

Thanks