Question:
Show that $f: R \rightarrow R$, given by $f(x)=x-[x]$, is neither one-one nor onto.
Solution:
We have, $f(x)=x-[x]$
Injection test:
$f(x)=0$ for all $x \in \mathbf{Z}$
So, f is a many-one function.
Surjection test:
Range $(f)=[0,1) \neq \mathbf{R}$.
So, f is an into function.
Therefore, f is neither one-one nor onto.
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