11At first glance, the constant function seems to be the only function, the conclusion follows from the fact that {f(x)}=0 for all x is the only soln to the above equation. I will give the solution only after verification.
Let [x] denote the greatest integer less than or equal to x and {x} = x-[x] (commonly known as fractional part of x).
Find all continuous functions f such that {f(x+y)} ={f(x)}+{f(y)}
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6 Answers
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71What if we take f(x) = x ?
eg, x= 2.4 , y = 1.3
{f(2.4+1.3)} = {3.7} = .7
{f(2.4)} + f(1.3)} = {2.4} + {1.3} = .7 , Hence looks like true if I'm not hurried.
11Vivek, use the fact that {x+y}≤{x}+{y}
The above eqn must hold true for all x and y
71Yes, the equation given by you certainly holds for all such values. But how does it support your statement that no function exists. Is my example wrong?
11Your example of f(x)=x looks wrong.
Okay here's my proof to get done with all confusions...
Observe that we can say {f(x)}≤12n for all integers n.
This is how I claim it:-
Put y=x to get {f(2x}=2{f(x)}.
Now 0≤{f(2x}<1 which gives 0≤{f(x)}<12
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Continuing like this for n steps we say 0≤{f(x)}<12n for all n, which eventually teaches us that {f(x)}=0
So it looks like f(x)=k for some integer k is the only soln.
