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Is the identity function always continuous?

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Question: Is the identity function $f:R\rightarrow R$ always continuous?

Answer: Yes.

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

If $f:X\rightarrow X$ is the identity function between two identical topological spaces, for any neighborhood $V$ of $f(a)$, we have $$f(V)=V\subset V.$$ $\blacksquare$