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If $f'$ is piecewise smooth on $[a,b]$, then $f$ is piecewise smooth on $[a,b]$.

Is this statement true?

I know that if we let $f'$ be piecewise smooth on $[a,b]$. Then, by definition, both $f'$ and $f''$ are piecewise continuous on $[a,b]$. Now, what I am left to show is that $\int f'$ is piecewise continuous on $[a,b]$.

In other words, is the anti-derivative of a piecewise continuous function also piecewise continuous?

Thanks in advance!

  • 1
    Assume $f$ is not continuous at some point $x$, what can you say about $f'$?2012-10-27

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