Q.Examine the differentiability of $f$, where $f$ is defined by $f(x) = \begin{cases} 1 + x, & x \le 2 \\ 5 - x, & x > 2 \end{cases}$ at $x = 2$.

Concept: Continuity Differentiability Relationship — a piecewise-linear function is continuous where its pieces meet in value, but is only differentiable there if the slopes on both sides also match.

Step 1 — Continuity: $\lim_{x\to2^-}f(x)=3$, $\lim_{x\to2^+}f(x)=3$, $f(2)=3$ — so $f$ is continuous at $x=2$.

Step 2 — Left-hand derivative: the piece $1+x$ has slope $1$, so $f'_-(2)=1$. …

Why This Is a Corner

$f$ is piecewise linear: slope $1$ on $x\le2$, slope $-1$ on $x>2$. Two different slopes meeting at a point always produce a corner, and a corner has no single tangent. Let's verify this formally.

Step 1 — Continuity at $x=2$

All three agree, so $f$ is continuous at $x=2$ — differentiability is still on the table.

Step 2 — Left-hand derivative

For $h<0$, $2+h\le2$ uses $f(2+h)=1+(2+h)=3+h$:

Step 3 — Right-hand derivative

For $h>0$, $2+h>2$ uses $f(2+h)=5-(2+h)=3-h$: …

Method: Checking Differentiability at a Corner of a Piecewise Linear Function

This method applies to piecewise functions made of two (or more) linear pieces meeting at a junction point, where you must determine whether the graph has a smooth tangent or a sharp corner there.

Steps

Step 1: Confirm continuity at the junction first

Compute the left-hand limit, right-hand limit, and the function's value at the junction; if all three agree, the function is continuous there, and differentiability remains a genuine possibility (though not yet guaranteed).

Step 2: Read off the slope of each linear piece

For a linear piece $f(x) = mx + c$, the derivative is simply the constant slope $m$ — no limit computation is strictly needed once the piece is recognised as linear, since $f'_-(a)$ equals the left piece's slope and $f'_+(a)$ equals the right piece's slope.

Step 3: Formally confirm using the definition, if required …

Common Mistakes

Mistake 1: Assuming continuity at the junction guarantees differentiability

Why it's wrong: many students verify the function is continuous at the point and stop there, treating that as sufficient proof of differentiability — but continuity only rules out a jump; it says nothing about whether the slopes on either side match. Correct approach: always compute and compare both one-sided derivatives separately, even after continuity has already been confirmed.

Mistake 2: Confusing the linear-piece slope shortcut with the general limit definition …