The double left continuous inverse is equal to the left continuous version of the function

I have some non-decreasing function $f$, and I will denote the left continuous inverse of $f$ as $f^{\leftarrow}(y)=inf\{x:f(x)\geq y\}$. Now I want to prove:$$(f^{\leftarrow})^{\leftarrow}(x)=f^{-}(x),$$where $f^{-}$ denotes the "left-continuous version" of $f$. I am thinking, that it is sufficient to prove, that they have the same points of continuity, but I have trouble proving it