11Vivek, use the fact that {x+y}≤{x}+{y}
The above eqn must hold true for all x and y
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
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.
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.
1171Yes, 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.
