Continuity of function $\lim_{t \rightarrow x} f(t)$ Let a function $f:A \rightarrow \mathbb{R}$, defined on a set $A\subset \mathbb{R}$ without isolated points, has a finite limit $g(x)$ in each point $x\in A$.
I wish to show that $g: A \rightarrow \mathbb{R}$, $g(x)=\lim_{t \rightarrow x} f(t)$, is continuous.  

Suppose $g$ is discontinuous at $a\in A$.  Let $a_k$ be a sequence in $A$ such that $d(a_k, a) < 1/k$  and such that for some constant $c>0$ and for all all $k$, $d(g(a_k),g(a))> c$. For each $k$ let $b_k$ have $d(b_k, a_k) < 1/k$ and $d(f(b_k),g(a_k)) < c/2$.  Then I claim that $b_k$ converges to $a$ yet $f(b_k)$ does not converge to $g(a)$.  This is a contradiction.
