Upper and Lower Derivatives of the Characteristic Function of Rationals?

[sbashrawi]

Homework Statement

What are the upper derivative and lower derivative of the characteristic function of rationals?

Homework Equations

The Attempt at a Solution

I think they are : upper derivative = 0
lower derivative = negative infinity

 

[micromass]

I agree that the lower derivative is 0. But I think the upper derivative depend on wether you take them in a rational or irrational point.

 

[zariuq]

If the point is rational: then the upper derivative is 0 and the lower derivative is negative infinity. (as you postulated)

However, if the point is irrational, then the upper derivative is infinity, and the lower derivative is 0. (for pretty much the same reason as in the rational case, except that the signs are reversed)