WBJEE 2025 Math Paper

The function
f(x)=x−[x]
is just the fractional-part function, usually denoted {x}=x−⌊x⌋.
On any open interval (n,n+1) (where n∈ℤ ), ⌊x⌋=n is constant, so
f(x)=x−n
is continuous there.
At an integer x=n, the left- and right-hand limits disagree:
As x⟶n−,
f(x)=x−(n−1)⟶n−(n−1)=1.
As x⟶n+,
f(x)=x−n⟶0
(and f(n)=0 ).
Hence each integer n is a jump discontinuity.
Answer: the points of discontinuity are exactly the integers, ℤ.