$\textbf{Statement:}$ Let $\Phi:(a,b)\to \mathbb{R}$ be a function.Then $\Phi$ is convex if and only if for each closed subinterval $[c,d]\subset (a,b)$ we have, \begin{equation} \Phi(x)=\Phi(c)+\int_{c}^{x}\phi(t)dt : c\leq x\leq d......(1), \end{equation} where $\phi:\mathbb{R}\to\mathbb{R}$ is monotone non-decreasing and left continuous function. Also, $\Phi$ has left and right derivatives at each point of $(a,b)$ and they are equal except perhaps for at most a countable number of points.
can someone explain what does left continuity of $\phi$ means and how to prove it?I am finding it difficult to visualize it geometrically.
Left continuity means
$$\lim_{x \to c^-}f(x) = f(c)$$
Or that we approach it from below. For example, $1_{>0}(x)$ is left continuous, but not right continuous at $x=0$