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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?

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You're right; the continuity assumption is not needed. Most likely, it is a "typo". – David Mitra Feb 23 '13 at 19:33
@DavidMitra Thanks a lot. That answers my question, so maybe you can post this as an answer? – Spenser Feb 23 '13 at 19:37
up vote 2 down vote accepted

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

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