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Say we have a piecewise continuous* function $f:\mathbb{R}→\mathbb{R}$. Consider $g(x)\equiv \int_a^x \frac{df(x')}{dx'}dx'$. The integral $g$ is continuous as far as I know from Riemann integration. So is $g$ the uniquely "stitched together" version of $f$?

The same question in other words: if I began with a continuous function $g$ and added distinct constants $c_i$ to it on intervals $[a_i,a_{i+1}]$, I would have $f$. Then does $\int_a^x \frac{df(x')}{dx'}dx'$ recover $g$?

*edit: I meant to imply differentiable, as Martin pointed out. I leave it unaltered for generality.

zahbaz
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If $f'$ exists and is integrable, yes and yes. $f$ and $f + $ a piecewise constant function will have the same derivative in almost everywhere point.

Martín-Blas Pérez Pinilla
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