I am studying real analysis with Bartle & Sherbert's Introduction to Real Analysis. There is the following theorem:
Interior Extremum Theorem: Let $c$ be an interior point of the interval $I$ at which $f:I\to\mathbb{R}$ has a relative extremum. If the derivative of $f$ at $c$ exists, then $f'(c)=0$.
The proof is easy to follow, and note that there is no mention about the continuity of $f$ on $I$. But then this corollary follows:
Corollary: Let $f:I\to\mathbb{R}$ be continuous on an interval $I$ and suppose that $f$ has a relative extremum at an interior point $c$ of $I$. Then either the derivative of $f$ at $c$ does not exist, or it is equal to zero.
It is certainly true, but why you have to make the further assumption that $f$ is continuous? Can't we say from the previous theorem that it is true for discontinuous functions as well?
Best Answer
You're right; the continuity assumption is not needed. Most likely, it is a "typo".