Any help with this discrete math problem as pictured aboveSo

Any help with this discrete math problem as pictured above?

Solution

Let, x=m+f

where m is the integer part and f is the fractional part.

Let, y=n+g be another real number

SO that:

f(x)=f(y)

Hence

(m+f)m=(n+g)n

m^2+mf=n^2+gn

m^2-n^2=gn-fm

(m-n)(m+n)=gn-fm

Taking absolute value of both sides

|(m-n)(m+n)|=|gn-fm|

g<1,f<1

Assume, m and n are not equal

Hence,

|(m-n)(m+n)|=|gn-fm|<|m-n| which is a contradiction

Hence, m=n

Hence f is injective.