A basic non-autonomous O.D.E question Consider the following non-autonomous O.D.E
$$ \dot{x}(t) = h(x(t),g(t))$$
such that $h(.,.)$ is continuous but $g(.)$ is discontinuous(step function). Does the solution exist here ? I don't think so.

It depends on what you call a solution. For example, if you are looking for a continuously differentiable function $x$, then there is no such solution. You can, however, consider a merely continuous solutions to the following integral equation
$$
x(t)=x_0+\int_{t_0}^t h(x(\tau),g(\tau))d\tau.
$$ 
Then it can be proved (pick up almost any textbook on optimal control) that there exists a unique solution. 
