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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.

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Student-friendly explanation

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.

How to write this in exams

  1. 1

    Start with the exact idea

    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. 2

    Then show how to use it

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

  3. 3

    Add one concrete 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.

  4. 4

    Avoid this incomplete answer

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

Definition

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

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.

Rule to remember

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.

Sequence to remember

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. A graph can show a sharp corner or cusp, but CBSE solutions should support the conclusion using LHD and RHD. Check continuity at the point. If continuous, compute LHD using the left branch and RHD using the right branch. Equate LHD and RHD. Use the continuity equation and derivative equation together to find constants

Memory hook

Differentiability is continuity plus equal one-sided slopes.

Examples and method

Worked example

Let f(x) = {x^2, x <= 1; ax + b, x > 1}. For differentiability at x = 1, continuity gives 1 = a + b. LHD = derivative of x^2 at 1 = 2. RHD = derivative of ax + b = a. Hence a = 2. From 1 = a + b, b = -1. Therefore f is differentiable at x = 1 when a = 2 and b = -1.

Method to apply

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

Diagram support

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

How CBSE asks it

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

Avoid common mistakes

Common confusion

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

Common wrong answer

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

Exam tip

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

Quick check

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

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

Answer writing and exam use

1-mark answer

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-mark answer

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

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. 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. Let f(x) = {x^2, x <= 1; ax + b, x > 1}. For differentiability at x = 1, continuity gives 1 = a + b. LHD = derivative of x^2 at 1 = 2. RHD = derivative of ax + b = a. Hence a = 2. From 1 = a + b, b = -1. Therefore f is differentiable at x = 1 when a = 2 and b = -1. Often asked as a parameter-finding problem where a piecewise function must be continuous and differentiable at the joining point. Differentiating both branches and equating slopes without first applying continuity, leading to incomplete parameter values.
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