Question

Question: a) Prove that a strictly increasing function from R to itself is one-to-one.

b) Give an example of an increasing function from R to itself is not one-to-one.

Short answer

Answer:

$\text{f(x)=0forx>0andf(x)=xfor}x⩽0$

Step-by-step solution

Step: 1

Let f be the given strictly increasing function from R to itself.

If $a<b$, then $f\left(a\right)>f\left(b\right)$if$a>b$ then$f\left(a\right)<f\left(b\right)$

Thus if $a\ne b$, then$f\left(a\right)\ne f\left(b\right)$

Step: 2

Therefore,

$f\left(x\right)=0$for $x>0$and $f\left(x\right)=x$for$x⩽0$