Show that a two variable function is positive everywhere, given it's positive on the x-axis and satisfies some extra conditions

Question

Let $f(x,t)$ from $\mathbb{R^2}$ to $\mathbb{R}$ be a continuously differentiable function such that $\frac{df(x,t)}{dt}= \frac{df(x,t)}{dx}$ with $0<f(x,0)$ for all $x$. Show that for all $x,t$ we have $0<f(x,t)$

I don't know how to go about solving this problem. My ideas go in the direction of using either some form of Greens theorem or maybe define a vectorfield with f in both of its entries (Since it would satisfy the integrability conditions).

Answer 1

A hint:

The function $f$ is constant along lines $x+t={\rm const.}$ Each such line intersects the $x$-axis in a point $(x,0)$ where we know that $f(x,0)>0$.