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?