Limit iff Limits from Left and Right

Theorem
Let $f$ be a real function defined on an open interval $\openint a b$ except possibly at a point $c \in \openint a b$.

Then:
 * $\map f x \to l$ as $x \to c$


 * $\map f x \to l$ as $x \to c^-$
 * $\map f x \to l$ as $x \to c^-$

and
 * $\map f x \to l$ as $x \to c^+$

Necessary Condition
Let $\map f x \to l$ as $x \to c$.

Then from the definition of the limit of a function:
 * $\forall \epsilon > 0: \exists \delta > 0: 0 < \size {x - c} < \delta \implies \size {\map f x - l} < \epsilon$

So for any given $\epsilon$, there exists a $\delta$ such that:
 * $0 < \size {x - c} < \delta$

implies that:
 * $l - \epsilon < \map f x < l + \epsilon$

Now:

That is: $\forall \epsilon > 0: \exists \delta > 0$:


 * $(1): \quad c - \delta < x < c \implies \norm {\map f x - l} < \epsilon$
 * $(2): \quad c < x < c + \delta \implies \norm {\map f x - l} < \epsilon$

So given that particular value of $\epsilon$, we can find a value of $\delta$ such that the conditions for both:


 * $(1): \quad f$ tending to the limit $l$ as $x$ tends to $c$ from the left

and :
 * $(2): \quad f$ tending to the limit $l$ as $x$ tends to $c$ from the right.

Thus:
 * $\displaystyle \lim_{x \mathop \to c} \map f x = l$

implies that:
 * $\displaystyle \lim_{x \mathop \to c^-} \map f x = l$

and:
 * $\displaystyle \lim_{x \mathop \to c^+} \map f x = l$

Sufficient Condition
Let $\map f x \to l$ as $x \to c^-$ and $\map f x \to l$ as $x \to c^+$.

This means that:
 * $(1): \quad \forall \epsilon > 0: \exists \delta > 0: c - \delta < x < c \implies \size {\map f x - l} < \epsilon$

and :
 * $(2): \quad \forall \epsilon > 0: \exists \delta > 0: c < x < c + \delta \implies \size {\map f x - l} < \epsilon$

In the same manner as above, the conditions on $\delta$ give us that:

So:
 * $\forall \epsilon > 0: \exists \delta > 0: 0 < \size {x - c} < \delta \implies \size {\map f x - l} < \epsilon$

Thus:
 * $\displaystyle \lim_{x \mathop \to c^-} \map f x = l$

and:
 * $\displaystyle \lim_{x \mathop \to c^+} \map f x = l$

together imply that:
 * $\displaystyle \lim_{x \mathop \to c} \map f x = l$

Hence the result.