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:

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

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