Composition of Continuous Functions

Given $f:A\to \mathbb{R}$ and $g:B\to \mathbb{R}$ suppose that the range $f(A)=\{f(x):x\in A\}$ is contained in the domain $B$ so that the composition $g\circ f(x)=g(f(x))$ is defined on $A$.

If $f$ is continuous at $c\in A$ and if $g$ is continuous at $f(c)\in B$ then $g\circ f$ is continuous at $c$.