Multiplicative Functional Equation

f(xy) = f(x)f(y)

In this video, I solve the following neat question: Find all continuous functions f such that f(xy) = f(x) f(y). One solution would be f(x) = x^2, but can you think of other ones? I solve this with a clever transformation, by reducing this equation to one that is more familiar. I also discuss the case where x and y are 0 and/or negative, and even mention the non-continuous case. Enjoy!

Other functional equations:

f(x+y) = f(x) + f(y):    • fx+y = fx + fy   (Cauchy's functional equation)

f(x+y) = f(x)f(y):    • f(x+y) = f(x)f(y)  

f(f(f(x))) = x:    • Press fff to pay respects  

f'(x) = f(f(x)):    • A new hard differential equation  

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