# Uniform Continuity Proof

• October 14th 2007, 11:11 PM
tbyou87
Uniform Continuity Proof
Prove that f is uniformly continuous on [1,2] by assuming the fact that f is continuous on (0,inf).

Since f is continuous (0, inf) then f is continuous on [1,2]. Then by theorem, If f is continuous on a closed interval [1,2], then f is unifromly continuous on [1,2].

I was just wondering how I can justify that if f is continuous on (0, inf) then f is continuous on [1,2]. It seems obvious just I don't know how it should be formally. Like a restriction of a continuous function is continuous.

Thanks
• October 15th 2007, 02:42 AM
Plato
Because $\left[ {1,2} \right] \subset \left( {0,\infty } \right)$ then if $x \in \left[ {1,2} \right]$, f is continous at x by the given.