Integration and existence of an antiderivative

Question

Let $ f: \mathbb {R} \rightarrow {\mathbb {R}} $ be given by

$$ f (x) = \left \{\begin {matrix} \sin ({\frac {1} {x}}), & \text {if} \phantom {a} x \neq 0; \\  c, & \text {if} \phantom {a} x = 0 \end {matrix} \right. $$ where $ c \in [-1,1]$. For what values ​​of $c$ is there an antiderivative of $f$?

I do not know of a theorem of the form "if and only if" that it tells me when a function $ f $ has no antiderivative.

Anyone have a suggestion?

Answer 1

Hint: Define $F(x) = \int_0^x \sin (1/t)\, dt.$ Then $F'(x) = \sin(1/x),$ $x\ne 0.$ Does $F'(0)$ exist?