ISC Class 12 Maths: Continuity and Differentiability Explained

Why continuity and differentiability matter?

Ever wondered why a roller‑coaster can be smooth in some parts and suddenly jolt? That’s continuity and differentiability playing out in real life, and they’re the backbone of many calculus problems you’ll face in ISC exams.

💡 In Simple Words: A function is continuous if you can draw its graph without lifting the pen. It’s differentiable if, at every point, you can place a straight line that just touches the curve – that line is the tangent.

What is continuity?

Think of water flowing through a pipe. If the pipe has a hole or a sudden jump, the flow stops or splashes. A continuous function is like a perfect pipe – the flow (the function’s values) never breaks.

Formal definition (first time explained)

We say a function f(x) is continuous at a point a if three things happen:

• The limit of f(x) as x approaches a exists. (A limit is the value the function gets closer to as x gets closer to a.)
• The function actually has a value at a – f(a) is defined.
• The limit you found equals the actual value: limx→af(x) = f(a).

How to check continuity at a point

Follow these three quick steps:

1. Find the left‑hand limit (approach from smaller x) and the right‑hand limit (approach from larger x). They must match.
2. Make sure the function is defined at that point.
3. Confirm the limit equals the function’s value.

What is differentiability?

Imagine you’re hiking up a hill. The steepness at any spot is the derivative. If the hill has a sharp corner, you can’t assign a single steepness – that’s a point where the function isn’t differentiable.

Formal definition (first time explained)

A function f(x) is differentiable at a if the limit

limh→0 [f(a+h) – f(a)] / h exists. This limit, if it exists, is the derivative f'(a). In plain words, we look at the slope of the line joining two points that get infinitesimally close.

Key point:

Differentiability automatically guarantees continuity, but the opposite isn’t always true.

Relation between continuity and differentiability

Worked example: checking continuity and differentiability

Consider the piecewise function:

f(x) = { x^2, if x ≤ 1
2x – 1, if x > 1 }

We’ll test at x = 1.

Step 1: Continuity

• Left‑hand limit: limx→1⁻ x^2 = 1.
• Right‑hand limit: limx→1⁺ (2x – 1) = 1.
• Function value: f(1) = 1^2 = 1.

All three match, so f is continuous at 1.

Step 2: Differentiability

• Derivative from left: d/dx (x^2) = 2x → 2·1 = 2.
• Derivative from right: d/dx (2x – 1) = 2 → constant 2.

Since both one‑sided derivatives equal 2, the derivative exists. Hence f is differentiable at 1.

Quick decision flow

📝 Likely Exam Questions

• Q1. State the definition of continuity at a point and give one example of a function that is continuous but not differentiable.
• Answer: A function f is continuous at a if limx→af(x) exists and equals f(a). Example: f(x)=|x| is continuous everywhere but not differentiable at x=0 because the left and right slopes differ.
• Q2. Verify whether f(x)=\begin{cases}x^2,&x\le2\\3x-2,&x>2\end{cases} is continuous and differentiable at x=2.
• Answer: Left limit =4, right limit =4, f(2)=4 ⇒ continuous. Left derivative =4, right derivative =3 ⇒ derivatives differ, so not differentiable at x=2.
• Q3. Explain why every differentiable function must be continuous, using the limit definition.
• Answer: If f'(a) exists, the limit of [f(a+h)-f(a)]/h as h→0 exists. Multiplying by h and taking h→0 shows f(a+h)→f(a), which is precisely the condition for continuity.
• Q4. Sketch the graph of a function that is continuous at x=0 but not differentiable there, and label the point.
• Answer: Graph y=|x|. The curve meets at (0,0) without a gap, but the corner makes the tangent undefined.