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Question

Show that the Signum function f:RR, given by

f(x)=1, if x>00, if x=01, if x<0
is neither one-one nor onto.

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Solution

f(x)=1ifx>00ifx=01ifx<0.

It is seen that f(1)=f(2)=1, but 12.

f is not one-one.
Now, as f(x) takes only 3 values (1,0, or1).
For the element 2 in co-domain R, there does not exist any x in domain R such that f(x)=2
f is not onto
Hence, the signum function is neither one-one nor onto.

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