- #1
robertjordan
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Homework Statement
Show there exists a function [itex]f: \mathbb{R} \rightarrow \mathbb{R}[/itex] s.t. [itex]f^2=f[/itex] but [itex]f\neq{0,1}[/itex].
Homework Equations
Here [itex]f^2=f[/itex] means for arbitrary [itex]a\in{\mathbb{R}}, f(a)^2=f(a)[/itex]
The Attempt at a Solution
I came up with the function [itex]f(a)= \begin{cases}
0, & \text{if }a\text{> 0 } \\
1, & \text{if }a \leq 0
\end{cases}[/itex]What do you guys think? Is this right? I figured the only real numbers r for which r^2=r are r=0 and r=1 so the function f will have to only spit out those values or else there would be some input a for which f(a)^2=/=f(a)
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