1. The problem statement, all variables and given/known data Prove that if f is uniformly continuous on [a,b] and on [a,c] implies that f is uniformly continuous on [a,c]. 2. Relevant equations 3. The attempt at a solution This is my rough idea for a proof, can someone help be say this more formally? Is my thinking even correct? Let epsilon > 0. Then there is some delta_1 for [a,b] and some delta_2 for [b,c]. Then the minimum of delta_1 and delta_2 is the delta we want for [a,c].