Continuity

• The precise meaning of continuity at a point and on an interval — NCERT Class 12 Chapter 5.
• To test continuity using left-hand limit, right-hand limit, and f(a) = lim x→a f(x).
• Types of discontinuity: removable, jump, and infinite — with graph interpretation.
• Algebra of continuous functions: sums, products, quotients (where defined), and composition.
• Continuity of polynomial, rational, trigonometric, exponential, and modulus functions on their domains.

Level 1 — Foundations

Verbal: A function f is continuous at x = a if the graph has no break, hole, or jump at that point — you can draw it without lifting the pencil.

Definition (three conditions at x = a):

1. f(a) is defined.
2. lim(x→a) f(x) exists (LHL = RHL).
3. lim(x→a) f(x) = f(a).

Continuity on [a, b]: Continuous at every interior point; at endpoints check one-sided limits.

Standard continuous functions: Polynomials everywhere; rational except where denominator = 0; sin x, cos x everywhere; eˣ, aˣ everywhere; log x for x > 0.

Level 2 — JEE / NEET depth

Discontinuity types:

Piecewise functions (JEE favourite): Define f(x) differently on intervals; check continuity at junction points by equating one-sided limits and function value.

Intermediate Value Property: If f is continuous on [a,b] and k lies between f(a) and f(b), then ∃ c ∈ (a,b) with f(c) = k. Used to prove existence of roots.

Modulus continuity: |f| is continuous wherever f is continuous. |x − a| is continuous everywhere.

Composition: If g is continuous at a and f is continuous at g(a), then fog is continuous at a.

Test continuity of a piecewise function at x = 0

f(x) = { x² sin(1/x),  x ≠ 0
               { 0,               x = 0

Step 1 — At x = 0: f(0) = 0 is defined.
Step 2 — For x ≠ 0: |x² sin(1/x)| ≤ x² → 0 as x → 0 (squeeze theorem).
Step 3 — lim(x→0) f(x) = 0 = f(0).
Step 4 — All three conditions satisfied → f is continuous at x = 0.

Locate discontinuity in a rational function

g(x) = (x² − 1)/(x − 1).

Step 1 — Domain: x ≠ 1 (denominator zero).
Step 2 — For x ≠ 1: g(x) = (x−1)(x+1)/(x−1) = x + 1.
Step 3 — lim(x→1) g(x) = 2 exists but g(1) is undefined.
Step 4 — **Removable discontinuity** at x = 1; define g(1) = 2 to make continuous.

• Is f(x) = |x − 3| continuous at x = 3?
• Classify discontinuity of 1/(x−2)² at x = 2.
• For f(x) = { ax+1, x≤1; 3, x>1 }, find a for continuity at x=1.
• Where is tan x discontinuous on ℝ?
• Stretch: Prove sum of two continuous functions is continuous using ε–δ or limit laws.

NCERT Chapter 5 link: Continuity precedes differentiability in NCERT ordering. Every differentiable function is continuous, but converse fails — prepare standard counterexample f(x) = |x| at x = 0 with graph sketch showing corner but no break.

Exam connections: Piecewise functions with unknown constants are CBSE staples — equate LHL, RHL, and f(a). JEE tests modulus compositions and trigonometric limits combined with continuity. Intermediate Value Theorem proves root existence without finding exact root — state theorem before concluding.

Study strategy: For piecewise definitions, circle junction points before computing anything. Log domain continuity requires x > 0; rational functions exclude zeros of denominator. Practice classifying discontinuity type with one-line justification each.

When studying Continuity within Calculus, start by listing every formula and definition on one page without looking at the textbook. Compare your list to NCERT — missing items indicate gaps to fix immediately. Work through at least two NCERT Examples for this section with steps written in full; examiners award method marks even when arithmetic slips.

For board exams (CBSE), long answers benefit from a clear structure: definition → explanation → diagram or formula → example → brief conclusion. Underline key terms. For JEE Main and NEET, prioritise conceptual traps and quick calculation paths; timed mixed quizzes of 10 questions after revision simulate exam pressure.

Cross-topic link: Coordinate geometry and vectors often combine with matrices; calculus links to physics kinematics problems.

Spaced revision: Review this note at 1 day, 3 days, and 7 days after first study. Attempt the Quick check questions closed-book, then open the Practice tab for graded reinforcement. Maintain an error log — repeated mistake patterns reveal whether the issue is concept, formula recall, or careless reading.

Diagram and terminology drill: For Mathematics, redraw key figures from memory and define every labelled part in one sentence. Vocabulary precision prevents mark loss in descriptive answers — use NCERT terms exactly as printed in the textbook.

Revision tip: Link this topic to adjacent Class 12 chapters before attempting mixed practice.

Open the Practice tab for graded questions on Continuity.