Continuity at lines through a point implying continuity at the point?

[uman]

Homework Statement

Let f:\mathbb{R}^2\to\mathbb{R} be continuous everywhere except, possibly, at the origin. Furthermore, for any point p\in\mathbb{R}^2, let s_p:\mathbb{R}\to\mathbb{R}^2 be defined by s_p(t) = tp. Now assume that f\circ s_p is continuous, as a function \mathbb{R}\to\mathbb{R}, for all p. Does this necessarily imply that f 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 f is uniquely 1. f 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 R^2\setminus \{0\} is not compact.

 

[LCKurtz]

See what you can do with something like

\vec R(r,\theta) = \langle r\cos\theta,r\sin\theta,\sin\theta\rangle,\ 0\le r \le 1,\ 0\le\theta\le 2\pi

 

[LCKurtz]

Don't know why it won't let me edit that post, but here's a better one:

<br /> \vec R(r,\theta) = \langle r\cos\theta,r\sin\theta,|\sin\theta|\rangle,\ 0\le r \le 1,\ 0\le\theta\le 2\pi<br />

Do a parametric plot of it and see what you think.