Compact functions w/ same zeros, then exists cont. fcn h : ℝ 4 → ℝ so f = gh or...

[ccm]

Compact functions w/ same zeros, then exists cont. fcn h : ℝ 4 → ℝ so f = gh or...

Problem: If f: ℝ⁴ →ℝ and g:ℝ⁴ → ℝ are two smooth functions with the same zeros, than it exists a continuous function h : ℝ ⁴ → ℝ so that f = g ⋅ h or g = h ⋅f.

Could you tell me how to start?

 

[HallsofIvy]

Look at \(\displaystyle \frac{f}{g}\) and \(\displaystyle \frac{g}{f}\). The only possible discontinuities are where g= 0, for the first, or where f= 0, for the second. Since f and g have the same zeros, those are all of the form "\(\displaystyle \frac{0}{0}\)". Can you show that, for at least one of those, the discontinuities are removable?