Let $B_t$ be any standard Brownian motion and let $f \in L^2_\text{loc}$. Then $W = \int_0^{\cdot} f(\cdot,s)\, dB_s$ is a continuous local martingale.
This is stated in my lecture notes right after the definition of 'local martingale' without a proof. I don't know if this is supposed to be trivial or if the proof is omitted as an exercise. Since this seems like a general result can somebody point be in a direction where I can find more information or a proof for this theorem?