First slide

Algebra of real valued functions

Question

If $f : R \to R$ is defined by $f (x) = x - [x] - \frac{1}{2}$ for $x \in R,$ where $[x]$ is the greatest integer not exceeding x, then $\{x \in R : f (x) = \frac{1}{2}\}$ =

Moderate

A

Z
B

N
C

$ϕ$
D

R

Solution

$0 \leq x - [x] < 1 \Rightarrow - \frac{1}{2} \leq x - [x] - \frac{1}{2} < \frac{1}{2}$
$\Rightarrow - \frac{1}{2} \leq f (x) < \frac{1}{2} \Rightarrow f (x) \neq \frac{1}{2} \forall x$
$\Rightarrow \{x \in R : f (x) = \frac{1}{2}\} = ϕ$

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