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Let $f$ be a continuous function on $\mathbb R^2\setminus \{0\}$. In a standard multivariable calculus course, one learns that even if $x\mapsto f(x,0)$ and $y\mapsto f(0,y)$ extend to be continuous at 0, $f$ need not do so. Indeed, it is not even sufficient to have $t\mapsto f(at, bt)$ continuous at 0 (with the same value there) for all $a,b\in \mathbb R$. This can be seen by taking

$$f(x,y) = \frac{x^2 - y^3}{x^2+y^3},$$

which satisfies $f(at, bt) \rightarrow 1$ as $t\rightarrow 0$ for any $a$ and $b$, but $f(t^{3/2}, t) \rightarrow 0$.

I am wondering if there is a sufficient condition on the limits of compositions of $f$ with one-variable functions which guarantees that $f$ extends to be continuous at 0. Is it sufficient to have $f(p(t), q(t))$ continuous at 0 (with the same value at 0) for all polynomials $p(t)$ and $q(t)$ which vanish at the origin? What about for all analytic functions?

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