Given any continuous function f(x), does there always exist a function whose integral is f(x) ?
For instance, is there a function whose integral is Weierstrass function
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No, any integral function of an $L^1$ function is absolutely continuous, while there exist continuous functions that aren't absolutely continuous. For example, the Weierstrass function, being not differentiable in any point, isn't of bounded variation (any function of bounded variation is differentiable almost everywhere) and then not absolutely continuous (because absolute continuity implies bounded variation). So the Weierstrass function is a continuous function that is not an integral function of any $L^1$ function.
If, when you write “integral”, what you mean is “primitive”, then the answer is negative, of course. Because if $F$ is a primitive of $f$ then, by definition, $F'=f$. In particular, $F$ is differentiable. But the Weierstrass function isn't.