differentiable functions are continuous

Proposition 1.

Suppose I is an open interval on R, and f:IC is differentiableMathworldPlanetmathPlanetmath at xI. Then f is continuous at x. Further, if f is differentiable on I, then f is continuous on I.


Suppose xI. Let us show that f(y)f(x), when yx. First, if yI is distinct to x, then


Thus, if f(x) is the derivative of f at x, we have

limyxf(x)-f(y) = limyxf(x)-f(y)x-y(x-y)
= limyxf(x)-f(y)x-ylimyx(x-y)
= f(x) 0
= 0,

where the second equality is justified since both limits on the second line exist. The second claim follows since f is continuous on I if and only if f is continuous at x for all xI. ∎

Title differentiable functions are continuous
Canonical name DifferentiableFunctionsAreContinuous
Date of creation 2013-03-22 14:35:27
Last modified on 2013-03-22 14:35:27
Owner matte (1858)
Last modified by matte (1858)
Numerical id 8
Author matte (1858)
Entry type Theorem
Classification msc 57R35
Classification msc 26A24
Related topic DifferentiableFunctionsAreContinuous2
Related topic LimitsOfNaturalLogarithm