You can't prove that "f is continuous". What you can prove is that "f is continuous at x=c".
Do you know what "f is continuous at x=c" is defined to mean?
Do you know what g(c) is? (And what's your justification for that?) What about the value of h(c)?
If so, then, given that that inequality (h(x)<=f(x)<=g(x)) holds in some neighborhood of x=c,
what can you say about f(c)?