Let $f$ be a function $[a,b] \to \mathbb{R}$. I define antiderivative as a function $F$ such that $F' = f$, and I define indefinite integral of $f$ as the function $\int_0^x f(t) dt$.
The Fundamental Theorem of Calculus [ Part 1 ] says that if $f$ is continuous, then the indefinite integral is an antiderivative.
However, a discontinuous $f$ can also have an antiderivative, such as the function $f(x) = \sin(1/x)$. See here for a proof not involving the indefinite integral. The accepted answer of this question proves that the indefinite integral is an antiderivative.
I am wondering, if a discontinuous function has an antiderivative, must it also be the indefinite integral up to a constant?
