  In this work, we discuss the continuity of $h$-convex functions by
introducing the concepts of $h$-convex curves ($h$-cord). Geometric
interpretation of $h$-convexity is given. The fact that for a $h$-continuous
function $f$, is being $h$-convex if and only if is $h$-midconvex is proved.
Generally, we prove that if $f$ is $h$-convex then $f$ is $h$-continuous. A
discussion regarding derivative characterization of $h$-convexity is also
proposed.
