Dividing both sides by a Dirac delta function - ok?

[pellman]

Suppose I wind up with the relation

$$f(x)\delta (x-x')=g(x)\delta (x-x')$$

true for all x'.

Can I safely conclude that f(x) = g(x) (for all x)? Or am I overlooking something? this is a little too close to dividing both sides by zero for comfort.

 

[CompuChip]

You could integrate your relation over any interval containing x, and use the definition of the delta function (or rather, distribution):

If
$$f(x)\delta (x-x')=g(x)\delta (x-x')$$
then
$$\int_{x - \epsilon}^{x + \epsilon} f(x)\delta (x-x') \, \mathrm{d}x' = \int_{x - \epsilon}^{x + \epsilon} g(x)\delta (x-x') \, \mathrm{d}x'$$
which evaluates (by definition of the delta) to
$$f(x) = g(x)$$

 

[pellman]

Thank you!