calculation – Two continuous functions $ f, g $

Suppose that $ f: (a, b) rightarrow mathbb {R} $ continuous function and $ g: (b, + infty) rightarrow mathbb {R} $ uniformly continuous function and $ f (b) = g (b). $ I have to prove that the function

$ h (x) = start {case} f (x) text {si} a the x the b \ g (x) text {si} x ge b end {case} $

is a uniformly continuous function. Honestly, I've been trying to solve this problem for some time. I've tried using the definition that since $ g $ is uniformly continuous then for $ delta> $ 0 and for all $ x, y $ we have $ | x-y | < delta $ involved $ | g (x) – g (y) | < epsilon $ but I can not use that definition for $ h $