When you think about this question, you need to think about g(x)  get close to g(a) thus making f(g(x)) close enough to f(g(a)).

For example,  think about a point with a ball. If x closes enough to a, the ball contain g(x) to g(a) will close to be a point. So that, another ball (w=g(x) close enough to g(a) ) is small enough that can be put into a ball f(x). Therefore, to get f(w) get arbitrarily close to f(g(a)). So it can provided f(g(x)) close enough to f(g(a)), f(g(X)) is continuous at a.