Does there exist a function $f: \mathbb{R^2} \to \mathbb{R}$ such that it has both partial derivatives in every point of $\mathbb{R^2}$ and $\lim_{t\to0}f(t, t^2) = \infty$?
2026-04-01 20:24:53.1775075093
Everywhere partially differentiable function satisfying $\lim_{t\to0}f(t, t^2) = \infty$
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I don't know about Nate Eldredge's function (edit: it was proposed $x^2$ instead of $x^4$ in the denominator), but $$f(x,y)=\frac{xy}{x^{4}+y^{2}}$$ with $f(0,0)=0$ sure works.
Ps. Sorry I can't comment other answers (not yet!).