Assume that $$ f$$ is continuous at $$ a$$, now we will prove that $$ \underset{h\to <!--\; \to \; -->0}{lim}f(a+h)=f\left(a\right)$$. To show this limit exists, we refer the the definition of limit of a function. So let $$ \mathrm{ϵ<!--\; ϵ\; -->}\in <!--\; \in \; -->{\mathbb{R}}^{><!--\; >\; -->0}$$, then since $$ f$$ is continuous at $$ a$$ then we know there is some $$ {\mathrm{\delta <!--\; \delta \; -->}}_c\in <!--\; \in \; -->{\mathbb{R}}^{><!--\; >\; -->0}$$ such that $$ \mathrm{\forall <!--\; \forall \; -->}x\in <!--\; \in \; -->dom\left(f\right),|x-<!--\; -\; -->a|<<!--\; <\; -->{\mathrm{\delta <!--\; \delta \; -->}}_c⟹<!--\; ⟹\; -->|f\left(x\right)-<!--\; -\; -->f\left(a\right)|<<!--\; <\; -->\mathrm{ϵ<!--\; ϵ\; -->}$$.

We need to prove that there is some $$ \mathrm{\delta <!--\; \delta \; -->}\in <!--\; \in \; -->{\mathbb{R}}^{><!--\; >\; -->0}$$ such that $$ \mathrm{\forall <!--\; \forall \; -->}h\in <!--\; \in \; -->dom\left(f\right),|h-<!--\; -\; -->0|<<!--\; <\; -->\mathrm{\delta <!--\; \delta \; -->}⟹<!--\; ⟹\; -->|f(a+h)-<!--\; -\; -->f\left(a\right)|<<!--\; <\; -->\mathrm{ϵ<!--\; ϵ\; -->}$$, so suppose that $$ h\in <!--\; \in \; -->\text{dom}\left(f\right)$$, thus we can also consider the value of $$ h+a\in <!--\; \in \; -->\text{dom}\left(f\right)$$. Thus by the assumption of continuity, as it holds for all $$ x$$ in $$ \text{dom}\left(f\right)$$ then $$ |(h+a)-<!--\; -\; -->a|<<!--\; <\; -->{\mathrm{\delta <!--\; \delta \; -->}}_c⟹<!--\; ⟹\; -->|f(a+h)-<!--\; -\; -->f\left(a\right)|<<!--\; <\; -->\mathrm{ϵ<!--\; ϵ\; -->}$$, therefore taking $$ \mathrm{\delta <!--\; \delta \; -->}={\mathrm{\delta <!--\; \delta \; -->}}_c$$ will complete this direction of the proof.

Now we work in the other direction, first assuming that $$ \underset{h\to <!--\; \to \; -->0}{lim}f(a+h)=f\left(a\right)$$ and trying to show that $$ f$$ is continuous at $$ a$$, so let $$ {\mathrm{ϵ<!--\; ϵ\; -->}}_c\in <!--\; \in \; -->{\mathbb{R}}^{><!--\; >\; -->0}$$, thus since the aformentioned limit exists, then we have some $$ \mathrm{\delta <!--\; \delta \; -->}\in <!--\; \in \; -->{\mathbb{R}}^{><!--\; >\; -->0}$$ such that $$ \mathrm{\forall <!--\; \forall \; -->}h\in <!--\; \in \; -->\text{dom}\left(f\right),|h-<!--\; -\; -->0|<<!--\; <\; -->\mathrm{\delta <!--\; \delta \; -->}⟹<!--\; ⟹\; -->|f(h+a)-<!--\; -\; -->f\left(a\right)|<<!--\; <\; -->{\mathrm{ϵ<!--\; ϵ\; -->}}_c$$.

Working in s similar fashion as in our first direction, we consider the value $$ x-<!--\; -\; -->a\in <!--\; \in \; -->\text{dom}\left(f\right)$$, since the latter statement in the previous paragraph is for all $$ h\in <!--\; \in \; -->\text{dom}\left(f\right)$$, we can consider $$ h=x-<!--\; -\; -->a$$, which says that $$ |x-<!--\; -\; -->a-<!--\; -\; -->0|<<!--\; <\; -->\mathrm{\delta <!--\; \delta \; -->}⟹<!--\; ⟹\; -->|f(x-<!--\; -\; -->a+a)-<!--\; -\; -->f\left(a\right)|<<!--\; <\; -->{\mathrm{ϵ<!--\; ϵ\; -->}}_c$$ which is equivalent to $$ |x-<!--\; -\; -->a|<<!--\; <\; -->\mathrm{\delta <!--\; \delta \; -->}⟹<!--\; ⟹\; -->|f\left(x\right)-<!--\; -\; -->f\left(a\right)|<<!--\; <\; -->{\mathrm{ϵ<!--\; ϵ\; -->}}_c$$, which is exactly what we wanted to prove.