# Math Help - Continuity on an interval

1. ## Continuity on an interval

Show that if f is continuous on [0,infinity) and uniformly continuous on [a,infinity) for some positive constant a, then f is uniformly continuous on [0,infinity)

2. Originally Posted by gixxer998
Show that if f is continuous on [0,infinity) and uniformly continuous on [a,infinity) for some positive constant a, then f is uniformly continuous on [0,infinity)
Since $f$ is continuous on $[0,a]$ it is uniformly continuous, so there exists $\delta_1$ such that $|x-y|<\delta_1\implies|f(x)-f(y)|<\epsilon$. Similarly, there exists $\delta_2$ for the interval $[a,\infty)$.

So taking $\delta=\min\{\delta_1,\delta_2\}$ will work for $[0,\infty)$.