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medium difficulty9 min

Continuity at a Point

A function f is continuous at x = a if f(a) is defined, lim(x → a) f(x) exists, and lim(x → a) f(x) = f(a).

In one line: Continuity needs three matches: left value, right value, and actual value.

Practice This Concept

Understand it

Continuity means there is no break in the function at the point being tested. In exam questions, check the left-hand limit, right-hand limit, and actual value of the function. For piecewise functions, the point where the rule changes is the main point to test.

Example

For f(x) = {kx + 1, x ≤ 2; 3x - 1, x > 2}, continuity at x = 2 requires 2k + 1 = 3(2) - 1 = 5, so k = 2.

Formula and rule

Continuity at x = a: LHL = RHL = f(a).

Algebra of continuous functions: if f and g are continuous at a, then f + g, f - g, fg, and f/g are continuous at a, provided g(a) ≠ 0.

See it worked out

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

Worked example

Let f(x) = {x + 3, x ≤ 1; px + q, 1 < x < 3; 2x - 1, x ≥ 3}.

  1. 1For continuity at x = 1: 1 + 3 = p + q, so p + q = 4.
  2. 2For continuity at x = 3: 3p + q = 2(3) - 1 = 5.
  3. 3Subtracting, 2p = 1, so p = 1/2.
  4. 4Then q = 4 - 1/2 = 7/2.
  5. 5Hence p = 1/2 and q = 7/2.

Method you can reuse

  1. 1Identify the point of possible discontinuity.
  2. 2Find the left-hand limit.
  3. 3Find the right-hand limit.
  4. 4Find f(a) using the rule that includes equality.
  5. 5Equate LHL, RHL, and f(a), then solve for the unknown constants.

Visual tip

A graph may help show a break, jump, or removable gap, but algebraic one-sided limit checking is enough for standard CBSE questions on this concept.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

If f(x) = x2 for x < 1 and f(x) = ax + 2 for x ≥ 1 is continuous at x = 1, find a.

Think first, then show answer
Left-hand limit = 1. Since f(1) = a + 2, continuity gives a + 2 = 1, so a = -1.

Common wrong answer

Using the expression for x > a to calculate f(a) when the function actually defines f(a) in the x ≤ a branch.

Don't mix this up

Students often equate only the two formula pieces and forget to compare them with f(a), especially when the equality sign belongs to one side of a piecewise function.

Exam tip

At a joining point, first decide which expression gives f(a), then compare it with the left-hand and right-hand limits.

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

Usually asked by giving a piecewise function and asking for unknown constants that make it continuous at one or two points.
  1. 1

    Enough for 1 mark

    A function f is continuous at x = a if f(a) is defined, lim(x → a) f(x) exists, and lim(x → a) f(x) = f(a).

  2. 2

    Add for 2 marks

    Example: For f(x) = {kx + 1, x ≤ 2; 3x - 1, x > 2}, continuity at x = 2 requires 2k + 1 = 3(2) - 1 = 5, so k = 2.

  3. 3

    Add for 3 marks

    Continuity means there is no break in the function at the point being tested.

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