View source for Smooth point of a function

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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.



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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]].