Let $I=[a,b]$ with $a<b$ and let $u:I\rightarrow\mathbb{R}$ be a function with bounded pointwise variation, i.e. $$Var_I u=\sup\{\sum_{i=1}^n|u(x_i)-u(x_{i-1})|\}<\infty$$
where the supremum is taken over all partition $P=\{a=x_0<x_1<...<x_{n-1}<b=x_n\}$. How can I prove that if $u$ satisfies the intermediate value theorem (IVT), then $u$ is continuous?
My try: $u$ can be written as a difference of two increasing functions $f_1,f_2$. I know that a increasing function that satisfies the (ITV) is continuous, hence, if I prove that $ f_1,f_2$ satisfies the (ITV) the assertion follows. But, is this true? I mean, $f_1,f_2$ satisfies (ITV)?
Begin by proving that every function of bounded variation has finite one-sided limits at every point. (Decomposition into monotone functions does this in one line.) Then observe that the intermediate value property fails unless $f(a+)=f(a-)=f(a)$.