Show that $f(t, B_t)$ is only a function about $t$.

Question

Let $B_t$ be a Brownian motion. For a two-variable smooth function $f(t, x)$ with second order differentiable continuous (i.e., $f\in C^{1,2}$), if $f(t, B_t)$ is a finite variation process, show that $f$ is only a function about $t$.

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My work:

I only try the Ito formula $$ df(t, B_t)=\frac{\partial f}{\partial t}dt+\frac{\partial f}{\partial B_t}dB_t+\frac{1}{2}\frac{\partial^2 f}{\partial B_t^2}dt $$

Answer 1

Since $f(t,B_t)$ is a finite variation process,its quadratic variation

$$ \left[f(s,B_s)\right]_t=\int_0^t \left(\frac{\partial }{\partial x}f(s,B_s)\right)^2 ds=0,\forall t >0 $$.This means $\frac{\partial f}{\partial x}=0$.Hence $f $ is only a function about $t$.