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The notion ''"Differentiability implies continuity as well as integrability"'' is at least misleading. You want to say - I suppose - that, if a function is differentiable (and therefore continuous) in a point (and therefore in a neighborhood of this point), then there exists a primitive for this function on this neighborhood.

The notion ''"Differentiability implies continuity as well as integrability"'' is at least misleading. You want to say - I suppose - that, if a function is differentiable (and therefore continuous) in a point (and therefore in a neighborhood of this point), then there exists a primitive for this function on this neighborhood.

Integrability of a function <math>f(x)</math> on <math>\mathbb{R}</math> OTOH is commonly understood to be  the existence of <math>\int_{\mathbb{R}}f(x) dx</math> - and the simple example of <math>f(x)= \sin(x)</math> shows that the Integral of this differentiable function does not exist.--[[User:DiEb|DiEb]] 17:01, 17 August 2008 (EDT)

Integrability of a function <math>f(x)</math> on <math>\mathbb{R}</math> OTOH is commonly understood to be  the existence of <math>\int_{\mathbb{R}}f(x) dx</math> - and the simple example of <math>f(x)= \sin(x)</math> shows that the Integral of this differentiable function does not exist.--[[User:DiEb|DiEb]] 17:01, 17 August 2008 (EDT)