Define the function $f:[0,1] \to \mathbb{R}$ by $ f(x)=\int_E x^tg(t)d\mu(t) $ where $E \subset \mathbb{R^+}$, $\mu$ is a nonnegative measure on $\mathbb{R}$ and $g:\mathbb{R} \to \mathbb{R}$ is a $\mu$-integrable function, that is $\int |g|d\mu < \infty$.
Is $f$ a continuous function of $x$?
I would be tempt to use the relation between absolute continuity and the lebesgue integral but as the measure is not Lebesgue, it's of no use.
Is it possible to show that $f$ is continuous?
Does this need any extra assumptions?