Limit of Constant Function/Two-Sided Limit at Real Number

Theorem

Let $a, b \in \R$.

Define $f : \R \to \R$ by:

$\map f x = a$ for each $x \in \R$.

Then:

$\ds \lim_{x \mathop \to b} \map f x = a$

where $\ds \lim_{x \mathop \to b}$ denotes the limit as $x \to b$.

We have:

$\size {\map f x - a} = 0$ for all $x \in \R$.

So for any $\epsilon > 0$ and $\delta > 0$, we have:

$\size {\map f x - a} < \epsilon$ for all $x \in \R$ with $\size {x - b} < \delta$.

So by the definition of the limit as $x \to b$, we have:

$\ds \lim_{x \mathop \to b} \map f x = a$

$\blacksquare$