Is a constant function always continuous?

Question: Is a constant function $f:R\rightarrow R$ always continuous?

Answer: Yes.

Why: For a given $\epsilon$ choose arbitrary $\delta$.

If $f:X\rightarrow Y$ is a constant function between two topological spaces, for any neighborhoods $U$ of $a$ and $V$ of $f(a)$, we have $$f(U)=\{f(a)\}\subset V.$$ $\blacksquare$

See Examples of maps.