Quddusaliquddus stop reading if you don't want the solution.(adsbygoogle = window.adsbygoogle || []).push({});

OK?

Stopped?

Right, to find all cont, functions from R to R satisfying f(x+y)f(x-y) = (f(x)f(y))^2

put x=y=0 to see f(0)^2=f(0)^4 ie f(0) =0 1 or -1

also x=y shows f(0)f(2x)=f(x)^4

so f(0)=0 implies f=0

if f(0)=1 then we see the relation

f(2x)=f(x)^4

if you're stupid you get mixed up and conclude that only the constant solution f=1 works. Then you realize that there are obviously non-constant solutions, duh! such as 2^{x^2} so you figure out why these ones are the only kind:

so, let f(1) = k

then f(2)=k^4, f(4)=k^16, hmm, f(3) can be got from f(3)f(1)=(f(2)f(1))^2

and pretty soon you realize that for every integer n, f(n)=k^{n^2}

now you go on to think, but we can do it for all rationals with powers of 2 in the denominator, and it works there, and as they're dense in the reals you see that actually f_k(x) = k^{x^2} forms a complete set of solutions (well, there's the solutions g_k = -f_k but whose counting).

**Physics Forums - The Fusion of Science and Community**

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Solution to quddusaliquddus's cont func. questions if you want it

Loading...

Similar Threads - Solution quddusaliquddus's cont | Date |
---|---|

I Finding the explicit solution of a trig equation | Nov 21, 2017 |

A Find positive integer solutions to a/(b+c)+b/(a+c)+c/(a+b)=4 | Aug 6, 2017 |

B Integer solutions for equations | Jul 6, 2017 |

B Solution set of a square root | May 8, 2017 |

Limits cont (x-2/x-3) = 1 + 1/(x-3) | Oct 8, 2011 |

**Physics Forums - The Fusion of Science and Community**