Right and left side properties of limsup

Question

let $f:\mathbb{R}\to\mathbb{R}$ be a real function such that limsups from the left and right are finite. Let $g(x):=\limsup_{t\to x+}f(t)$. Is it true that $\limsup_{s\to x-}f(s)=\limsup_{s\to x-}g(s)$?