Unimodality and continuity for probability distribution From Wikipedia about the conditions for the Vysochanskij–Petunin inequality

The sole restriction on the distribution is that it be unimodal and
  have finite variance. (This implies that it is a continuous
  probability distribution except at the mode, which may have a non-zero
  probability.)

Does it mean that unimodality and finite variance imply the distribution is continuous except at the mode? Why is that? 
Thanks!

The finiteness of the variance is irrelevant. As explained by Sellke and Sellke (see the reference on the WP page on unimodality), in this context, the unimodality of a CDF $F$ with mode $m$ means that $F$ is concave on $[m,+\infty)$ and convex on $(-\infty,m]$. This is an easy exercise to show that any CDF which is unimodal in this sense has no jumps except possibly at $m$. 
Assuming for example that $F(x-)\lt F(x)$ at some $x\gt m$, one sees that for small values of $\varepsilon\gt0$, the segment between the points on the graph of $F$ with abscissae $x-\varepsilon$ and $x$ is at least partly above the graph. This contradicts the concavity hence $F(x-)=F(x)$ at every $x\gt m$.
In particular, discrete distributions $(p_k)$ on the integers which are unimodal in the sense that $k\mapsto p_k$ is increasing then decreasing, are not unimodal in this convexity/concavity sense.
