Prove there is an increasing function on closed bounded interval that is continuous only at points in $[a,b] \setminus C$

For the case $[a,b],$ the function $f$ is continuous at $a,$ but it may be that $a\in C.$

If $a\not\in C $ let $g=f$.

If $a\in C $ let $g(a)=-1$ and $g(x)=f(x)$ for $x\in (a,b]. $

Then $g$ is continuous only on $[a,b]$ \ $C.$