On approximately monotone and approximately H\"older functions

A real valued function $f$ defined on a real open interval $I$ is called $\mathrm{\Phi }$-monotone if, for all $x,y\in I$ with $x\le y$ it satisfies $$f(x)\le f(y)+\mathrm{\Phi }(y-x),$$ where $\mathrm{\Phi }:[0,\mathrm{\ell }(I)[\to {\mathbb{R}}_{+}$ is a given nonnegative error function, where $\mathrm{\ell }(I)$ denotes the length of the interval $I$. If $f$ and $-f$ are simultaneously $\mathrm{\Phi }$-monotone, then $f$ is said to be a $\mathrm{\Phi }$-H"older function. In the main results of the paper, we describe structural properties of these function classes, determine the error function which is the most optimal one. We show that optimal error functions for $\mathrm{\Phi }$-monotonicity and $\mathrm{\Phi }$-H"older property must be subadditive and absolutely subadditive, respectively. Then we offer a precise formula for the lower and upper $\mathrm{\Phi }$-monotone and $\mathrm{\Phi }$-H"older envelopes. We also introduce a generalization of the classical notion of total variation and we prove an extension of the Jordan Decomposition Theorem known for functions of bounded total variations.