# Let I = <mo stretchy="false">[ a , b <mo stretchy="false">] with a &l

Let $I=\left[a,b\right]$ with $a and let $u:I\to \mathbb{R}$ be a function with bounded pointwise variation, i.e.
$Va{r}_{I}u=sup\left\{\sum _{i=1}^{n}|u\left({x}_{i}\right)-u\left({x}_{i-1}\right)|\right\}<\mathrm{\infty }$
where the supremum is taken over all partition $P=\left\{a={x}_{0}<{x}_{1}<...<{x}_{n-1}. 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)?
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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 $f\left(a+\right)=f\left(a-\right)=f\left(a\right)$.