function differentiable at only one point

Let f: be the function

f(x)={x,when x is rational,-x,when x is irrational.

See this entry ( Let g: be the function


Then g differentiableMathworldPlanetmathPlanetmath at 0, but everywhere else non-differentiable.

Indeed, since

g(0) = limh0f(h)h-f(0)0h
= limh0f(h)
= 0

g is differentiable at 0. If g would be continuous at x0, then f(x)=g(x)/x would be continuous at x. This result ( implies that g is non-differentiable away from the origin.

Title function differentiable at only one point
Canonical name FunctionDifferentiableAtOnlyOnePoint
Date of creation 2013-03-22 15:48:16
Last modified on 2013-03-22 15:48:16
Owner matte (1858)
Last modified by matte (1858)
Numerical id 6
Author matte (1858)
Entry type Example
Classification msc 57R35
Classification msc 26A24
Related topic FunctionContinuousAtOnlyOnePoint