Uniform Continuity Proof 2

Let f be a function on (a,b). Suppose that f is uniformly continuous on (a,c] and [c,b) for some c such that a<c<b. Then, Prove that f is uniformly continuous on (a,b) by using the epsilon - delta definition.

My first problem is that I don't get if i need to have epsilon1 and epsilon2 or delta1 or delta2 for each uniformly continuous function or if i can call them both by the same variable. I'm assuming you need to combine both definitions somehow to get it over the bigger interval.

Thanks