Can we find a function f:Rn→R that such that f is continuous and ∂vf(p) exists for all p∈Rn and v∈Rn. But f is not differentiable at 0?
Is such function f exists?
Here give a example that has directional derivative everywhere, but it's not continuous at the origin.
Answer
Consider the polar coordinates (r Φ) in R2. Function f:R2→R, which in polar coordinates is given by:
f(r Φ) := r⋅sin(3⋅Φ)
is continuous everywhere, is infinitely differentiable outside the origin O, and f has all directional derivative at O, but f is not differentiable at O.
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