Determining All Continuous Functions s.t.

[quddusaliquddus]

I want to determine all continuous functions s.t. for all x, y reals:

f(x+y)f(x-y) = {f(x)f(y)}^2

Now, I want to know where on the web I can learn how I go about doing this, because I don't know what methods to use or what I should be aiming at in manipulating the above.

I don't want an answer - - I just want to know how to go-about if I want to do this...rather like how I know how to calculate the two roots of an equation, I'd go about factorising for example.

Thank you!

 

[matt grime]

I realize you don't want solutions just hints, but it's always a fine line. I hope I don't cross it (I certainly won't give you the answer in full because right now I don't know it). But, have you actually considered what the function can be like? In particular, have you considered any special values of x and y? This isn't even using the continuity property (though I am presuming you mean functions from R to R), you'll find there are actually a lot of things you can prove just considering special cases.

 

[matt grime]

Right, I think I have figured out the answer, and the thing you need to know about is Heine's criterion of continuity: a function is continuous at x iff for all sequences x_n tending to x, f(x_n) tends to f(x)

 

[quddusaliquddus]

Cheers. As always - you can trust matt to give you his time. :)

 

[matt grime]

I'm just avoiding writing my own maths down, that's all; and this is a less guilty way to 'waste' time (ie pretend it's productive) than playing golf, and counts as a reasonable use of academic resources (free permanently on broadband connection on my desk).

And I haven't solved it as it turns out - I got something the wrong way round.

Let us know how far you've got.

 

[matt grime]

Ok, now I've got it, honest, guv. Heine isn't that useful. But knowing that a continuous function is determined by its values on a dense subset is good (and guessing what the answer might be helps as well).

 

[Muzza]

matt, are there any non-constant solutions?

 

[matt grime]

Yep, there are, at least I'm fairly sure - the constraints mean I can't write them out here for people with more patience than me to check my working. There are, if this isnt' giving the game away too much, an uncountable number of solutions.

 

[quddusaliquddus]

The thing is I actually have the solution. But I don't understand how I could have come up with the answer. I don't knwo what prompted the person to follow the course of calculations that he did. Do you want to see the whole stuff i got, or would that be spoiling the fun?

 

[matt grime]

It's your question to do with as you please. If you wish I can give you the observations that I made (including the mistake) that let's you figure out the solutions.