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WBJEE 2025 Math Paper

Question : 17 of 75

Marks: +1, -0

f(x)=x-\lfloor x \rfloor, x \in \mathbb{R}

Solution:

The function

f(x)=x-\lfloor x \rfloor

is just the fractional-part function, usually denoted

\{x\}=x-\lfloor x \rfloor

On any open interval

(n, n+1)

(where

n \in \mathbb{Z}

\lfloor x \rfloor = n

is constant, so

f(x)=x-n

is continuous there.

At an integer

x=n

, the left- and right-hand limits disagree:

x \rightarrow n^{-}

f(x)=x-(n-1) \rightarrow n-(n-1)=1.

x \rightarrow n^{+}

f(x)=x-n \rightarrow 0

(and

f(n)=0

Hence each integer

n

is a jump discontinuity.

Answer: the points of discontinuity are exactly the integers,

\mathbb{Z}

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