Question

If the function $f:R\to R$ defined by $f\left(x\right)=\left[x\right]$, where $\left[x\right]$ is the greatest integer not exceeding $x$, for $x\in R$, then $f$ is:

• A. Even
• B. Odd
• C. Neither even nor odd
• D. Strictly increasing

Answer 1

The correct option is C

Neither even nor odd

Explanation for correct option

We can identify the nature of the given function by eliminating the options.

According to the options, we need to evaluate whether the function is even or odd.

We know that a function is said to be even if $f(-x)=f\left(x\right)$.

The given function $f\left(x\right)=\left[x\right]$ is not differentiable when $x\in I$ (integers), so the function is not even.

Similarly, for the function to be odd, the required condition is $f(-x)=-f\left(x\right)$, which will never be satisfied as the given function is not differentiable.

So, the function isn't odd either.

Hence, the given function is neither even nor odd.

Therefore, the correct option is (C) i.e. Neither even nor odd