On uniform deviation from the space of continuous function

Abstract

It's proven that for the normal space $X$ and any function $f\!:X\!\rightarrow\!\mathbb{R}$ the uniform distance $d(f,C(X))$ of function $f$ from the space $C(X)$ of all continuous functions $g\!:X\!\rightarrow \!\mathbb{R}$ is equal to the half of the uniform norm $\|\omega_{f}\|$ of the function's $f$ oscillation $\omega_{f}$ and it is reached on some function $g$ from $C(X)$