Show that this defines a continuous local martingale

Question

Let $f \in C^{1,2}([0, \infty) \times \mathbb{R})$ with $f_t = -\frac{1}{2}f_{xx}$. My notes state that from Itô's Lemma it follows that $X_t := f(t,B_t)$ defines a continuous local martingale (where $B_t$ is a standard brownian motion). I am not sure how this follows. I know that from Itô's Lemma we have: \begin{equation} X_t = f(0,0)+\int_0^tf_x(s,B_s)~dBs \end{equation} so it would be enough to show that $\int_0^tf_x(s,B_s)~dBs$ is a continuous local martingale. I would understand it if $f_x \in \mathcal{L^2_{loc}}$ but why should this be true here?