Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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?

share|cite|improve this question

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.