Define to be the radiation at angle from whatever starting point you want. Define to be . Then . So either f is identically 0, in which case for all or there exist [tex]\theta[tex] such that is positive and is negative. Since is continuous, so is and there must be some such that
What if we considered the function ? It follows from a fairly fundamental theorem about functions that is continuous on . We must then be in one of three cases:
- g(0) = 0
- g(0) < 0
- g(0) > 0
In the first case, we have and we are done.
Now, for the other two cases we must be more clever. We must first realise that and then apply the intermediate value theorem to . Do you think you can finish this off?
Edit: I was writing my answer at the same time as another user. I'll leave it here anyway, it might be useful.