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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?