1. ## Directional Derivative

Quick Question;

In Munkres' "Analysis on Manifolds", he defines the directional derivative of a map f at the point x, with respect to the vector u to be:

$\displaystyle \lim_{t \rightarrow 0} \frac{f(x + tu) - f(x)}{t}$, provided this limit exists.

Consider the function f(x,y) = |x| + |y|. $\displaystyle \lim_{t \rightarrow 0^{+}} \frac{f(x+tu) - f(x)}{t} \neq \lim_{t \rightarrow 0^{-}} \frac{f(x + tu) - f(x)}{t}$ and so the directional derivative at (0,0) with respect to u doesn't exist, according to the definition above (where it requires that both one-sided limits be equal to eachother). However, I have someone who is more knowledgeable in maths than me that says that it is only required that the limit exist when t approaches 0 through positive values. Am I right, or is he right?

EDIT: Even Wikipedia says that it is only required t go to 0 through positive values. But it just doesn't seem right that Munkres would make a mistake in his definition.

EDIT 2: http://mathworld.wolfram.com/DirectionalDerivative.html agrees with Munkres' definition.

I guess the definition depends on the author of the book?

2. Oops, misread and spoke too soon.