Constant and Identity Functions
Finally! Let’s evaluate some limits.
Constant Functions
Starting off with an easy one, we want to prove that for any constants $c \in \mathbb{R}$, we have
\[\lim_{x \rightarrow a} c = c\]Notice that
\[\lvert c - c \rvert = 0\]Therefore, the statement
\[\forall \epsilon > 0 \quad \exists \delta > 0 \quad \text{s.t.} \quad 0 < \lvert x - a \rvert < \delta \implies \lvert c - c \rvert < \epsilon\]is trivially true. Since
\[(\text{anything}) \implies (\texttt{true})\]is a tautology. Therefore, constant functions are continuous.
Identity Functions
Another simple proof, we want to show that
\[\lim_{x \rightarrow a} x = a\]Fix any $\epsilon > 0$ and let $\delta = \epsilon$, then
\[\exists \delta > 0 \quad \text{s.t.} \quad 0 < \lvert x - a \rvert < \delta \implies \lvert x - a \rvert < \epsilon\]which proves the result. Therefore, the identity function is continuous.