A function $f:D\to \mathbb R$ is continuous in $a\in D$ $\iff f$ is left and right continuous in $a$.
I firstly thought just to write down the definitions of left and right continuous and then it trivially shows the theorem. But apparently it isn't sufficient.
Let $f:D\to\mathbb R$ and consider $a\in D$. The function $f$ is rightcontinuous in $a \iff$ $$(\forall\epsilon\gt 0)(\exists\delta\gt 0)(a\le x\lt a+\delta\Rightarrow |f(x)-f(a)|\lt\epsilon)$$
and left continous $\iff$ $a \iff$ $$(\forall\epsilon\gt 0)(\exists\delta\gt 0)(a -\delta\lt x\le a\Rightarrow |f(x)-f(a)|\lt\epsilon)$$
So I found a proof online on this webpage.
My question is there another way to prove this maybe with the use of my definitions? I'd most appreciate it.