Proof that a Function is Uniformly Continuous

Let A and B be intervals. Suppose we have two uniformly continuous functions $f:A\to\mathbb{R}$ and $g:B\to\mathbb{R}$ such that for all $x\in{A\cap{B}}$, $f(x)=g(x)$.
Also suppose that $A\cap{B}$ is non-empty.
Define $h:A\cup{B}\to\mathbb{R}$ to be $f(x)$ if $x\in{A}$ and $g(x)$ if $x\in{B\setminus{A}}$.
Prove $h(x)$ is uniformly continuous.

My Attempt: I know that if the intersection is non-empty then for all points that are in the intersection $f$ and $g$ are uniformly continuous . So by epsilon delta definition we have $$(\forall\epsilon>0)(\exists\delta_1>0)(\forall{x,y}\in{A})(|x-y|<\delta_1\implies|f(x)-f(y)|<\epsilon),$$ and $$(\exists\delta_2>0)(\forall{x,y}\in{B})(|x-y|<\delta_2\implies|g(x)-g(y)|<\epsilon).$$ I was thinking maybe I could use these two definitions to show that $h$ is uniformly continuous on the set ${B}$ but from there I really don't know where to go. It seems intuitive to me that this must be true but I just don't know how to connect all the dots. Thanks for any help.