Difference between revisions of "Smooth point of a function"
From Encyclopedia of Mathematics
(Importing text file) |
(TeX) |
||
| Line 1: | Line 1: | ||
| − | An argument | + | {{TEX|done}} |
| + | 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. | 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. | ||
| Line 8: | Line 9: | ||
====Comments==== | ====Comments==== | ||
| − | Notice that any odd function, continuous or not, has | + | 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|Smooth function]]. | See also [[Smooth function|Smooth function]]. | ||
Latest revision as of 06:04, 22 August 2014
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
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