Differentiable multivariable functions

[hholzer]

Defining differentiability for multivariable functions we want not only
for the partial derivatives to exist but also local linearity.

Because my question is the same also for the single variable
case, I'll pose it with a single variable function.

In one variable we have that local linearity at a point is
such that E(x) = f(x) - L(x) tends to zero faster than
|x - a| as x -> a does.

My question is how can the difference between function values
tend to zero faster than the distance between |x-a|? My current,
and broken intuition, expects them to tend to zero at the same
time. Why is this not the case?

 

[Tinyboss]

Consider the function $$f(x)=x^2$$. How close to f(0)=0 is f(x), when x is 0.5? 0.1? 0.001?

 

[quasar987]

if the derivative is f'(a)>0, then it means that the ratio (f(x)-f(x))/(x-a) "stabilizes" to f'(a) as x-->a.. so wouldn't you say this implies that f(x)-f(x) and x-a to go 0 "at the same rate"?

 

[hholzer]

Tinyboss: I understand where you are going with that but why is that the case? What is the reasoning?

Quasar987: I would agree with that but my book is saying that |x-a| as x->a goes to zero faster than f(x) - L(x). Which is different than what you are saying.