Q.Every continuous function is differentiable. Examine whether this statement is true.

The modulus function $\phi(x) = |x|$ is continuous everywhere, including at $x = 0$. But at $x = 0$ the one-sided difference quotients differ:

$\lim_{h\to 0^{+}}\frac{|h|}{h} = 1,\qquad \lim_{h\to 0^{-}}\frac{|h|}{h} = -1,$ …

Step 1 — Note why the claim is tempting.

Many familiar continuous functions are also differentiable everywhere, for example

Each is continuous for all $x$ and differentiable for all $x$, which might suggest that continuity always forces differentiability.

Step 2 — Choose a candidate counterexample.

Consider the modulus function

It is continuous everywhere, including at $x = 0$, since $\lim\limits_{x \to 0} |x| = 0 = \phi(0)$.

Step 3 — Test differentiability at $x = 0$.

Examine the difference quotient from each side:

$\text{Left derivative: } \lim_{h \to 0^{-}} \frac{|0+h| - |0|}{h} = \lim_{h \to 0^{-}} \frac{-h}{h} = -1.$ …

Method: Testing a "Does Property A Imply Property B" Claim with a Known Counterexample

Use this whenever asked to examine whether one property (continuity, etc.) forces a stronger related property (differentiability, etc.). The general technique: know the true implication, know why the converse can fail, and keep a standard counterexample ready.

Steps

Step 1: Identify which direction of implication is actually a theorem

For continuity and differentiability, the proven direction is: differentiable at a point $\Rightarrow$ continuous at that point (provable directly from the definition of the derivative as a limit). Know this direction is solid before questioning the reverse.

Step 2: Recognise the claim under examination is the (unproven) converse

"Every continuous function is differentiable" reverses the arrow of the real theorem. A converse of a true implication is not automatically true — it must be checked independently, usually by trying to find where it fails.

Step 3: Recall or construct a function that is continuous but has a "corner" …

Common Mistakes

Mistake 1: Confusing the claim with the (true) reverse implication

Reasoning "differentiable functions are continuous, so the statement is true" conflates "every continuous function is differentiable" with the actual theorem "every differentiable function is continuous." Why it's wrong: these are different statements; only the second is a proven theorem, and confusing them leads to affirming a false converse. Correct approach: always identify precisely which direction of implication is being asked about before deciding whether it's provably true or needs a counterexample.

Mistake 2: Sign errors in the one-sided derivative limits …