In the dissertation, we described the structural properties of four function classes, determined the error function which is the most optimal one. Then we succeeded to show that the optimal error functions for approximate-monotonicity and approximate-Hölder property must be subadditive and absolutely subadditive, respectively. While the optimal error functions for approximate-convexity and approximate-affinity must possesses a special property, termed as Gamma-property. Then we offered a precise formula for the lower and upper approximately-monotone and approximately-Hölder envelopes. We also derive a formula for the approximately-convex minorant. We introduced a generalization of the classical notion of total variation and we proved an extension of the Jordan Decomposition Theorem known for functions of bounded total variations. Besides, we deduced Ostrowski- and Hermite-Hadamard-type inequalities from the approximate-monotonicity and approximate-Hölder properties, and then we also verified the sharpness of these implications. Using the notions of upper and lower interpolations, we established a characterizations for approximately monotone and approximately-Hölder functions. Finally, we characterized approximately-convex and approximately-affine classes of functions and investigated their relationship with approximately monotone and approximately Hölder functions.