Let $f$ be absolutely continuous on $[0,1]$ and thus $f'$ exists a.e. Suppose we have $f(x)=f'(x)$ a.e. Is it true that $f(x)=Ce^x$ a.e.?
Since $f$ is defined only a.e., can we use the usual integration as in an elementary calculus course to get $f(x)=Ce^x$ a.e.?
$f(x)e^{-x}$ is absolutely continuous and its derivative vanishes a.e.. Hence it is almost everywhere constant.