The function $f(x) = [x]$, where $[x]$ denotes the greatest integer function, is continuous at $x =$

(A) 2.9

(B) 5

(C) -3

(D) 6.5

Choose the correct answer from the options given below:

The greatest integer function $[x]$ gives the largest integer that is less than or equal to $x$.

$[2.9] = 2$

$[5] = 5$

$[-3] = -3$

$[6.5] = 6$

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The greatest integer function $[x]$ is continuous at all non-integer values and discontinuous at all integer values.

At integer values, the function "jumps" to the next value, creating a discontinuity.

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Checking each option:

(A) $2.9$ is not an integer, so $f(x)$ is continuous at $x = 2.9$

(B) $5$ is an integer, so $f(x)$ is discontinuous at $x = 5$

(C) $-3$ is an integer, so $f(x)$ is discontinuous at $x = -3$

(D) $6.5$ is not an integer, so $f(x)$ is continuous at $x = 6.5$

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At $x = 2.9$:

For values near $2.9$, such as $2.89$, $2.9$, and $2.91$:

$[2.89] = 2$

$[2.9] = 2$

$[2.91] = 2$

No jump occurs, so the function is continuous.

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At $x = 5$:

For values near $5$, such as $4.99$, $5$, and $5.01$:

$[4.99] = 4$

$[5] = 5$

$[5.01] = 5$

A jump from $4$ to $5$ occurs, so the function is discontinuous.

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The function is continuous at $(A) \, 2.9$ and $(D) \, 6.5$ only.