As I was a student, I got the following problem:
Let $f$ be a mapping from $[a,b]$ to $\mathbb{R}$ satisfying $(1)\quad f((1-\lambda)x+\lambda y)\leq (1-\lambda )f(x)+\lambda f(y)$ for all $x,y\in [a,b]$ and $\lambda\in [0,1]$. Suppose $f$ is differentiable on $]a,b[$. Show that if $f$ is continuous and f' increasing then $(1)$ holds and vice versa.
As a part of the solution, I fixed $y$, assumed $x
Now I thought that $\lim_{z\to x}\frac{f(z)-f(x)}{z-x}\leq \lim_{z\to x}\frac{f(y)-f(z)}{y-z}.$
My advisor said it is wrong and it should be
$\lim_{z\to x+}\frac{f(z)-f(x)}{z-x}\leq \lim_{z\to x+}\frac{f(y)-f(z)}{y-z}.$
I never got a proper reasoning why the formula I wrote is wrong. I assumed that $x\leq z$ so I thought $\lim_{z\to x}$ means $\lim_{z\to x+}$ as $z$ can't approach $x$ from left. Could anyone explain me why limit does not mean one sided limit if I can't compute the two-sided limit?