Q.Examine the differentiability of $f$, where $f$ is defined by $f(x) = \begin{cases} x[x], & 0 \le x < 2 \\ (x - 1)x, & 2 \le x < 3 \end{cases}$ at $x = 2$ (here $[x]$ denotes the greatest integer function).

Concept: Continuity Differentiability Relationship — differentiability requires continuity plus equal one-sided derivatives; here continuity holds but the one-sided derivatives disagree.

Step 1 — Continuity at $x = 2$: $\lim_{x \to 2^-} f(x) = 2$, $\lim_{x \to 2^+} f(x) = 2$, and $f(2) = 2$ — so $f$ is continuous at $x=2$.

Step 2 — Left-hand derivative: for $x \in [1,2)$, $f(x) = x[x] = x$, so $f'_-(2) = 1$. …

Setting Up

The formula for $f$ changes at $x=2$, so that is the only point to examine. For a piecewise function, always check continuity first, then compare the one-sided derivatives.

For $1 \le x < 2$, $[x]=1$, so the first piece is $f(x)=x[x]=x$. For $2 \le x < 3$, $f(x)=(x-1)x$, so $f(2)=(2-1)(2)=2$.

Step 1 — Continuity at $x=2$

Both one-sided limits equal $f(2)=2$, so $f$ is continuous at $x=2$ — differentiability is still possible.

Step 2 — Left-hand derivative

For $h<0$ small, $2+h\in(1,2)$, so $f(2+h)=2+h$:

Step 3 — Right-hand derivative

For $h>0$ small, $2+h\in(2,3)$, so $f(2+h)=(1+h)(2+h)=2+3h+h^2$: …

Method: Differentiability of a Piecewise Function Involving the Greatest Integer Function

This method applies to piecewise functions where one branch contains $[x]$ (the greatest integer / floor function), and differentiability must be examined at the exact junction point between two branches.

Steps

Step 1: Determine the constant value of $[x]$ just to the left of the junction

The greatest integer function is constant on each interval between consecutive integers, so identify which integer $[x]$ equals for $x$ slightly less than the junction point — this may differ from $[x]$ AT the junction point itself.

Step 2: Check continuity first, since it is a necessary condition

Compute the left-hand limit, right-hand limit, and the function's own value at the junction (using whichever piece's condition actually includes that point), and confirm all three agree.

Step 3: Compute the left-hand derivative using the correct constant value of $[x]$

$f'_-(a) = \lim_{h \to 0^-} \frac{f(a+h) - f(a)}{h}$ …

Common Mistakes

Mistake 1: Using the wrong value of $[x]$ near the junction point

Why it's wrong: it's tempting to plug the junction point's own integer value into $[x]$ (e.g. using $[2]=2$) when computing the left-hand derivative, but the left-hand limit approaches from $x$ values strictly less than the junction, where $[x]$ takes the previous integer value instead. Correct approach: always determine $[x]$'s constant value on the open interval immediately to the left of the junction, not at the junction point itself.

Mistake 2: Concluding differentiability from continuity alone …