Q.The function $f:R\to Z$ defined by $f(x)=[x]$; where $[\,.\,]$ denotes the greatest integer function, is
(A) Continuous at $x=2.5$ but not differentiable at $x=2.5$
(B) Not Continuous at $x=2.5$ but differentiable at $x=2.5$
(C) Not Continuous at $x=2.5$ and not differentiable at $x=2.5$
(D) Continuous as well as differentiable at $x=2.5$

Concept: Greatest Integer Function — it jumps at every integer, but is constant (hence continuous and differentiable) on every open interval between integers.

Reasoning:

1. At $x = 2.5$, the function $f(x) = [x]$ equals $2$ for all $x$ in $(2, 3)$.
2. Since $2.5$ is not an integer, there is no jump at this point — the left-hand limit, right-hand limit, and $f(2.5)$ all equal $2$. …

The greatest integer function, $[x]$, returns the largest integer less than or equal to $x$. For any non-integer point, the function is locally constant — it doesn't jump there. The only trouble spots are the integers themselves, where the floor "steps up" by 1.

At $x = 2.5$, we are safely between 2 and 3. Let's check continuity and differentiability step by step.

1. Check continuity at $x = 2.5$

   For $x$ in the open interval $(2, 3)$, $[x] = 2$ exactly. So near $x = 2.5$, the function is the constant function $f(x) = 2$.

   The left-hand limit: $\lim_{x \to 2.5^-} f(x) = 2$.

   The right-hand limit: $\lim_{x \to 2.5^+} f(x) = 2$.

   The function value: $f(2.5) = [2.5] = 2$.

   Since the limit equals the function value, $f$ is continuous at $x = 2.5$.

2. Check differentiability at $x = 2.5$

   Differentiability requires the derivative to exist, i.e., the limit

must exist and be finite.

For any sufficiently small $h$ (say $|h| < 0.5$), $2.5 + h$ still lies in $(2,3)$, so $f(2.5 + h) = 2$.

Hence the difference quotient is

for all such $h \neq 0$. The limit as $h \to 0$ is clearly $0$.

Therefore $f'(2.5) = 0$, and the function is differentiable at $x = 2.5$. …

Method: Testing Continuity and Differentiability of the Greatest Integer Function at a Given Point

Use this method whenever asked about continuity/differentiability of $f(x)=[x]$ (or a similar step function) at a specific value.

Steps

Step 1: Check whether the given point is an integer

This is the single branching decision the whole method hinges on. Non-integers and integers behave completely differently for this function.

Step 2: If the point is NOT an integer, find its surrounding interval

For $x_0$ strictly between two consecutive integers $n$ and $n+1$, the function is constant, $f(x)=n$, throughout the open interval $(n, n+1)$ — there is no jump anywhere inside this interval.

Step 3: Test continuity using the local constant value

Since $f$ is locally constant near $x_0$, the left-hand limit, right-hand limit, and $f(x_0)$ are all trivially equal to $n$, so continuity holds automatically.

Step 4: Test differentiability using the difference quotient …

Common Mistakes

Mistake 1: Assuming the greatest integer function is never differentiable

Because $[x]$ is famous for its jumps, some students conclude it can't be differentiated anywhere. Why it's wrong: away from the integers, the function is locally constant, and a constant function is both continuous and differentiable (with derivative $0$) — the jumps are confined strictly to integer points. Correct approach: always separate "what happens at integers" from "what happens between integers" before answering.

Mistake 2: Treating a non-integer point as if it needs a jump/one-sided-limit analysis …