# 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 $$h$$