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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.


Solution

f:RR,f(x)=1, if x>00, if x=01, if x<0
It is seen that f(1)=f(2)=1 but 12. Therefore, f is not one-one.
Now, as f(x) takes only three values (1,0 or -1)for the element -2 in co-domain R, there does not exist any x in domain R such that f(x)=-2.
Therefore, f is not onto.
Hence, the Signum function is neither one-one nor onto. 

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