medium difficulty8 min

Differentiability and Continuity

A function f is differentiable at x = a if its left-hand derivative and right-hand derivative at a both exist and are equal.

In one line: Differentiability is continuity plus equal one-sided slopes.

Practice This Concept

Understand it

Differentiability checks whether the graph has a unique tangent slope at a point. A differentiable function must be continuous there, but a continuous function may fail to be differentiable because of a corner, cusp, vertical tangent, or unequal one-sided derivatives.

Example

For f(x) = |x|, f is continuous at x = 0, but LHD = -1 and RHD = 1, so f is not differentiable at x = 0.

Formula and rule

LHD = lim(h → 0-) [f(a + h) - f(a)]/h and RHD = lim(h → 0+) [f(a + h) - f(a)]/h.

Differentiability at a requires LHD = RHD.

Also, differentiability at a implies continuity at a.

See it worked out

Follow the example line by line, then use the same method on new questions.

Worked example

Let f(x) = {x², x ≤ 1; ax + b, x > 1}.

1For differentiability at x = 1, continuity gives 1 = a + b.
2LHD = derivative of x² at 1 = 2.
3RHD = derivative of ax + b = a.
4Hence a = 2.
5From 1 = a + b, b = -1.
6Therefore f is differentiable at x = 1 when a = 2 and b = -1.

Method you can reuse

1Check continuity at the point.
2If continuous, compute LHD using the left branch and RHD using the right branch.
3Equate LHD and RHD.
4Use the continuity equation and derivative equation together to find constants.

Visual tip

A graph can show a sharp corner or cusp, but CBSE solutions should support the conclusion using LHD and RHD.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

Is f(x) = |x - 2| differentiable at x = 2?

Think first, then show answer

No. LHD at x = 2 is -1 and RHD is 1, so the derivatives are unequal.

Common wrong answer

Differentiating both branches and equating slopes without first applying continuity, leading to incomplete parameter values.

Don't mix this up

A frequent wrong claim is that continuity automatically proves differentiability. It only proves that the first necessary condition is satisfied.

Exam tip

Before testing differentiability of a piecewise function, first check continuity. If continuity fails, differentiability fails immediately.

Write it in the exam

Each point is worth about one mark. Write point 1 for a 1-mark question and keep adding points as the marks go up.

How CBSE asks it

Often asked as a parameter-finding problem where a piecewise function must be continuous and differentiable at the joining point.

1
Enough for 1 mark
A function f is differentiable at x = a if its left-hand derivative and right-hand derivative at a both exist and are equal.
2
Add for 2 marks
Example: For f(x) = |x|, f is continuous at x = 0, but LHD = -1 and RHD = 1, so f is not differentiable at x = 0.
3
Add for 3 marks
Differentiability checks whether the graph has a unique tangent slope at a point.

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