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# Is a constant function always continuous?

From Mathematics Is A Science

Jump to navigationJump to search**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.