perplexus.info :: Just Math : Function Finding From Floor

Function Finding From Floor (Posted on 2016-09-30)

Find all functions f:R -> R such that:
f(floor(x)*y) = f(x)*floor(f(y)) holds for all x and y.

Prove that no other function satisfies the above relationship.

See The Solution Submitted by K Sengupta

Rating: 4.0000 (2 votes)

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Solution (spoiler)

| Comment 1 of 2

Let y = 0.

Then f(0) = f(x)*floor(f(0))

This means that f(x) must be a constant. Call it c.

Then, for all x and y,

c = c*floor(c)

Either c = 0 or floor(c) = 1.

So c can be 0 or c can be in [1,2) (i,e, >=1 and <2)

Posted by Steve Herman on 2016-10-01 12:45:56

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