Can we prove a right continuous real-function has left limits everywhere

Question

càdlàg function is basically a function defined on the real numbers (or a subset) that is everywhere right-continuous and has left limits everywhere.
Can right-continuity be the sufficient condition for a function to be a càdlàg function? In other words can we derive left limit on domain from right continuity?

Answer 1

NO.

Consider the function $$ f(x)=\left\{ \begin{array}{ccc} \sin(1/x) & \text{if} & x<0, \\ 0 & \text{if} & x\ge 0. \end{array} \right. $$ Then $f$ is right continuous everywhere, but has no left limit at $x=0$.