- #1
uman
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Homework Statement
Let [tex]f:\mathbb{R}^2\to\mathbb{R}[/tex] be continuous everywhere except, possibly, at the origin. Furthermore, for any point [tex]p\in\mathbb{R}^2[/tex], let [tex]s_p:\mathbb{R}\to\mathbb{R}^2[/tex] be defined by [tex]s_p(t) = tp[/tex]. Now assume that [tex]f\circ s_p[/tex] is continuous, as a function [tex]\mathbb{R}\to\mathbb{R}[/tex], for all [tex]p[/tex]. Does this necessarily imply that [tex]f[/tex] is continuous at the origin?
Homework Equations
None that I know of.
The Attempt at a Solution
I can envision some sort of spiral approaching the origin on which [tex]f[/tex] is uniquely 1. [tex]f[/tex] would be uniquely 0 sufficiently far away from the spiral, and the transition from 1 to 0 would be steeper and steeper as the spiral approaches the origin. I can't turn this into a formal counterexample but it makes me believe intuitively that finding a counterexample is possible. I wish I could give a better description of what I have in mind. :D
In other words, to find a counterexample, it should (in my intuition) be necessary to make use of the fact that [tex]R^2\setminus \{0\}[/tex] is not compact.