TITLE: Continuous extension of a function
QUESTION [6 upvotes]: Can anybody help me with this problem?

Justify whether the following statement is true or false:
Every continuous function on $\Bbb Q\cap [0,1]$ can be extended to a continuous function on $[0,1]$.

Any help will be appreciated.

REPLY [3 votes]: To add to Brian's answer, if $(X,d)$ is a metric space and $D$ is dense in $X$ and $f : D \rightarrow Y$ is a uniformly continuous function where $Y$ is a complete, metric space, then $f$ has a unique extension to $X$ and the extended function is also uniformly continuous.
In particular, in your case, every uniformly continuous function can be extended to a uniformly continuous function on the reals.