Year 12 Maths Advanced (2027) Applications of calculus

Differentiability & Continuity

20 practice questions 0 video lessons Theory + worked examples

NSW · Year 12 Mathematics Advanced · Applications of calculus

Differentiability and continuity ask whether a function is unbroken and smooth: a function must be continuous to be differentiable, but not every continuous function is differentiable.

Part of the NSW Year 12 Mathematics Advanced course, in the Calculus area of study (Applications of calculus focus area) of the 2024 syllabus. Work through practice questions with fully worked solutions and video lessons, or scroll down for the theory summary and worked examples.

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Practice questions

Every question with a fully worked solution.

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Theory

A function is differentiable only where its graph is smooth. This Year 12 Mathematics Advanced topic (MAV-12-06) distinguishes continuity (no breaks) from differentiability (a single tangent).

Continuous at \(x=a\) means no jump, hole or break. Differentiable means a single well-defined tangent exists there.

Differentiable \(\Rightarrow\) continuous, but not the reverse: corners, cusps, vertical tangents and any discontinuity are not differentiable.

\(y=|x|\) is continuous but not differentiable at the corner.

A useful implication:

\[\text{differentiable at }a\ \Rightarrow\ \text{continuous at }a\]

differentiable implies continuous, but not the reverse

Method

Check continuity: can you draw through the point without lifting the pen?
Check smoothness: is there one tangent (no corner or cusp)?
Conclude: smooth and unbroken means differentiable.

Example 1 — A corner

Is \(y=|x|\) differentiable at \(x=0\)?

Solution

Continuous, but the slope jumps from \(-1\) to \(+1\) — no single tangent, so not differentiable.

Example 2 — A smooth curve

Is \(y=x^{2}\) differentiable at \(x=0\)?

Solution

\(y'\)

\(=\)

\(2x=0\)

Smooth, so yes.

Example 3 — A jump

A graph jumps from \(1\) to \(3\) at \(x=2\). Continuous there?

Solution

The sides don't meet — a jump discontinuity, so not continuous (or differentiable).

Example 4 — Reasoning

A function is differentiable at \(x=5\). Is it continuous there?

Solution

Yes — differentiable \(\Rightarrow\) continuous.

Common pitfalls

Continuous is not differentiable. \(|x|\) is a continuous corner.

A break kills both.

Smooth means differentiable.

Frequently asked questions

What is the difference between continuous and differentiable?

Continuous means the graph has no breaks. Differentiable means it is also smooth, with a single tangent line. Differentiable functions are continuous, but not all continuous functions are differentiable.

Why is y = mod x not differentiable at 0?

Because it has a corner there: the slope is minus 1 on the left and plus 1 on the right, so there is no single tangent.

Does differentiable imply continuous?

Yes. If a function is differentiable at a point it must be continuous there.

Where is a function not differentiable?

At corners, cusps, vertical tangents, and any point where it is not continuous (a jump or hole).

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The First Derivative & the Sign of f'(x)