Q.Prove that the greatest integer function defined by $f(x) = [x]$, $0 < x < 3$, is not differentiable at $x = 1$ and $x = 2$.

A function that is not continuous at a point cannot be differentiable there. The greatest integer function $f(x)=[x]$ jumps at each integer.

At $x=1$: $\lim_{x\to 1^-}[x]=0$ but $\lim_{x\to 1^+}[x]=1$, so the limit does not exist and $f$ is discontinuous — hence not differentiable. Checking derivatives with $f(1)=1$: the right-hand derivative $\lim_{h\to 0^+}\dfrac{[1+h]-1}{h}=\lim_{h\to 0^+}\dfrac{0}{h}=0$, while the left-hand derivative $\lim_{h\to 0^-}\dfrac{[1+h]-1}{h}=\lim_{h\to 0^-}\dfrac{-1}{h}\to+\infty$; the …

On $0<x<3$ the greatest integer function is a staircase: $[x]=0$ on $(0,1)$, $[x]=1$ on $[1,2)$, $[x]=2$ on $[2,3)$. Differentiability requires continuity first, and where the graph jumps it cannot be continuous.

Discontinuity forces non-differentiability at $x=1$

Left: for $x$ just below $1$, $[x]=0$, so $\lim_{x\to 1^-}[x]=0$.

Right: for $x$ just above $1$, $[x]=1$, so $\lim_{x\to 1^+}[x]=1$.

The one-sided limits differ, so $\lim_{x\to 1}[x]$ does not exist — $f$ is discontinuous, hence not differentiable at $x=1$.

Confirm with the derivative definition at $x=1$

Using $f(1)=[1]=1$:

The right-hand derivative is $0$ and the left-hand derivative diverges, so they are unequal — $f'(1)$ does not exist. …

Method: Showing Non-Differentiability at a Jump Discontinuity

Use this method for functions like the greatest integer (floor) function that jump abruptly at certain points — this route is shorter than the corner-point method above because differentiability can be ruled out immediately once a jump is shown.

Steps

Step 1: Recall that differentiability requires continuity first

If a function is not even continuous at a point, it cannot be differentiable there — there is no need to compute a derivative limit at all. This shortcut saves work whenever a jump can be shown directly.

Step 2: Compute the left-hand and right-hand limits of the function itself (not yet the derivative) at the point in question

For the greatest integer function $[x]$ at an integer $n$, evaluate $[x]$ for $x$ slightly less than $n$ and slightly greater than $n$ separately.

Step 3: Compare the two one-sided limits to the function's actual value

If $\lim_{x\to n^-}f(x) \ne \lim_{x\to n^+}f(x)$, the two-sided limit does not exist, so $f$ is discontinuous at $n$ — and, by Step 1's logic, therefore automatically not differentiable there. …

Common Mistakes

Mistake 1: Trying to prove non-differentiability directly from the derivative limit without first checking continuity

Why it's wrong: it's more work, and easy to make sign errors, to jump straight into computing $\lim_{h\to0}\frac{[n+h]-n}{h}$ from both sides without first noticing the simpler fact that $[x]$ isn't even continuous at $n$. Correct approach: always check continuity first at a suspected trouble point — if it fails, non-differentiability follows immediately and no derivative computation is required.

Mistake 2: Evaluating $[1+h]$ or $[2+h]$ incorrectly for negative $h$

Why it's wrong: for $h$ a small negative number, $1+h$ is just below $1$ (e.g. $0.99$), so $[1+h]=0$, not $1$ — students sometimes assume the floor value doesn't change until $h$ crosses a whole unit. Correct approach: pick a concrete small value (like $h=-0.01$) and evaluate $[1+h]$ numerically before generalizing. …