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:
, provided this limit exists.
Consider the function f(x,y) = |x| + |y|. 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?
Oops, misread and spoke too soon.