Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be defined as $f(x) = [x]$, where [x] denotes the greatest integer less than or equal to x. Then which of the following statements are correct?

(A) f is one-one but not onto

(B) f is not onto

(C) f is not one-one

(D) f is one-one and onto

Choose the correct answer from the options given below:

The function $f(x) = [x]$ is the greatest integer function (or floor function), where $[x]$ gives the largest integer that is less than or equal to $x$.

Examples:

$[2.7] = 2$

$[5] = 5$

$[-1.3] = -2$

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Checking if $f$ is one-one:

A function is one-one if different inputs always give different outputs.

Take $x_1 = 1.2$:

$f(1.2) = [1.2] = 1$

Take $x_2 = 1.7$:

$f(1.7) = [1.7] = 1$

Here $x_1 \neq x_2$ but $f(x_1) = f(x_2)$.

Therefore, $f$ is not one-one.

Statement (C) is correct.

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Checking if $f$ is onto:

A function $f: \mathbb{R} \to \mathbb{R}$ is onto if every real number is an output.

The output of $[x]$ is always an integer: $..., -2, -1, 0, 1, 2, ...$

For example, there is no $x$ where $[x] = 1.5$.

Therefore, $f$ is not onto.

Statement (B) is correct.

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Since $f$ is not one-one, statement (A) is incorrect.

Since $f$ is neither one-one nor onto, statement (D) is incorrect.

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The correct statements are (B) and (C).