Interior Extremum Theorem and discontinuous functions. 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?

You're right; the continuity assumption is not needed. Most likely, it is a "typo".