# Smooth point of a function

From Encyclopedia of Mathematics

An argument $x$ of a function $f$ that satisfies the condition

$$\lim_{|h|\to0}\frac{|f(x+h)+f(x-h)-2f(x)|}{|h|}=0.$$

A point of differentiability of a function is a smooth point; generally speaking, the converse is not true. If a one-sided derivative exists at a smooth point, an ordinary derivative exists as well.

#### Comments

Notice that any odd function, continuous or not, has $x=0$ as a smooth point. For an additive function $f$ (i.e. $f(x+y)=f(x)+f(y)$ for all $x,y$), all points are smooth.

See also Smooth function.

**How to Cite This Entry:**

Smooth point of a function.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Smooth_point_of_a_function&oldid=33067

This article was adapted from an original article by V.F. Emel'yanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article