JEE Mains Continuity: Top 10 Problem Types You MUST Know
Master the essential continuity patterns that appear in every JEE Main paper. Save time and secure 4-6 marks with confidence.
Why Continuity Matters in JEE Mains
Continuity questions are consistently present in JEE Main papers, typically carrying 4 marks. Based on analysis of 2014-2024 papers, these 10 problem types cover 95% of all continuity questions asked.
🎯 Continuity Definition (JEE Perspective)
A function $f(x)$ is continuous at $x = a$ if:
This means: LHL = RHL = Functional Value at the point
🚀 Quick Navigation
Type 1: Piecewise Functions at Junction Points
Check continuity of $f(x) = \begin{cases} x^2 + 1 & \text{if } x \leq 2 \\ 4x - 3 & \text{if } x > 2 \end{cases}$ at $x = 2$
Standard Approach:
$\lim_{x \to 2^-} f(x) = \lim_{x \to 2^-} (x^2 + 1) = 2^2 + 1 = 5$
$\lim_{x \to 2^+} f(x) = \lim_{x \to 2^+} (4x - 3) = 4(2) - 3 = 5$
$f(2) = 2^2 + 1 = 5$ (using first case as x ≤ 2)
LHL = RHL = f(2) = 5 ✓
Conclusion: Continuous at x = 2
💡 JEE Tip:
Always check both sides separately for piecewise functions. The "equal to" case determines f(a).
Type 2: Modulus Functions
Check continuity of $f(x) = |x - 3|$ at $x = 3$
Solution:
$f(x) = \begin{cases} 3 - x & \text{if } x < 3 \\ x - 3 & \text{if } x \geq 3 \end{cases}$
$\lim_{x \to 3^-} f(x) = \lim_{x \to 3^-} (3 - x) = 0$
$\lim_{x \to 3^+} f(x) = \lim_{x \to 3^+} (x - 3) = 0$
$f(3) = |3 - 3| = 0$
💡 Key Insight:
Modulus functions are always continuous at the point where the expression inside becomes zero.
Type 3: Greatest Integer Function [x]
Check continuity of $f(x) = [x] + [x + 1]$ at integer points
Solution:
$f(n) = [n] + [n + 1] = n + n = 2n$
$\lim_{x \to n^-} f(x) = \lim_{h \to 0} [n - h] + [n - h + 1]$
$= (n - 1) + n = 2n - 1$
$\lim_{x \to n^+} f(x) = \lim_{h \to 0} [n + h] + [n + h + 1]$
$= n + (n + 1) = 2n + 1$
LHL = $2n - 1$, RHL = $2n + 1$, f(n) = $2n$
All three are different!
💡 Remember:
Greatest integer function [x] is discontinuous at all integers. LHL = n-1, f(n) = n, RHL = n.
Type 4: Trigonometric Functions with Parameters
Find the value of $k$ for which $f(x) = \begin{cases} \frac{\sin 2x}{x} & \text{if } x \neq 0 \\ k & \text{if } x = 0 \end{cases}$ is continuous at x = 0
Solution:
$\lim_{x \to 0} f(x) = \lim_{x \to 0} \frac{\sin 2x}{x}$
$\lim_{x \to 0} \frac{\sin 2x}{x} = 2 \cdot \lim_{x \to 0} \frac{\sin 2x}{2x} = 2 \cdot 1 = 2$
$\lim_{x \to 0} f(x) = f(0) \Rightarrow 2 = k$
💡 Standard Limits:
Memorize: $\lim_{x \to 0} \frac{\sin x}{x} = 1$, $\lim_{x \to 0} \frac{1 - \cos x}{x} = 0$, $\lim_{x \to 0} \frac{\tan x}{x} = 1$
Type 5: Exponential Functions with Parameters
Find $a$ and $b$ if $f(x) = \begin{cases} \frac{e^{ax} - e^{bx}}{x} & \text{if } x \neq 0 \\ 1 & \text{if } x = 0 \end{cases}$ is continuous at x = 0
Solution:
$\lim_{x \to 0} \frac{e^{ax} - e^{bx}}{x}$
$e^{ax} - e^{bx} = (1 + ax + \frac{a^2x^2}{2} + \cdots) - (1 + bx + \frac{b^2x^2}{2} + \cdots)$
$= (a - b)x + \frac{(a^2 - b^2)x^2}{2} + \cdots$
$\lim_{x \to 0} \frac{e^{ax} - e^{bx}}{x} = \lim_{x \to 0} \left[(a - b) + \frac{(a^2 - b^2)x}{2} + \cdots\right] = a - b$
$a - b = 1$
💡 Alternative Method:
Use L'Hôpital's Rule: $\lim_{x \to 0} \frac{e^{ax} - e^{bx}}{x} = \lim_{x \to 0} \frac{ae^{ax} - be^{bx}}{1} = a - b$
📚 Quick Continuity Checklist
Always Continuous:
- Polynomials
- Sine, Cosine functions
- Exponential functions
- Modulus functions (except composition issues)
Check Continuity At:
- Piecewise function junctions
- Points where denominator = 0
- Points inside modulus/greatest integer
- Parameter-dependent definitions
Problems 6-10 Available in Full Version
Includes 5 more essential JEE Main continuity problems:
- Rational Functions with Holes
- Composite Functions
- Functions with Parameters
- Continuity on Intervals
- Mixed Type Problems
⚠️ Common JEE Continuity Mistakes
Mistake 1: Forgetting to check both LHL and RHL separately
Always calculate $\lim_{x \to a^-} f(x)$ and $\lim_{x \to a^+} f(x)$ separately for piecewise functions.
Mistake 2: Wrong functional value at junction points
Use the case that includes "equal to" for f(a) in piecewise functions.
Mistake 3: Not using standard limits for trig functions
Memorize $\lim_{x \to 0} \frac{\sin x}{x} = 1$ and related limits to save time.
📝 Quick Self-Test
Test your understanding with these JEE-style problems:
1. Check continuity of $f(x) = \begin{cases} \frac{x^2 - 4}{x - 2} & x \neq 2 \\ 4 & x = 2 \end{cases}$ at x = 2
2. Find k if $f(x) = \begin{cases} \frac{1 - \cos 4x}{x^2} & x \neq 0 \\ k & x = 0 \end{cases}$ is continuous at x = 0
3. Check continuity of $f(x) = x - [x]$ at x = 1
🎯 JEE Exam Strategy
Time Management:
- Continuity problems: 3-4 minutes each
- Easy ones first: Modulus, Polynomials
- Save time on standard limits
- Skip and return if stuck
Scoring Strategy:
- 4 marks guaranteed if concepts clear
- Partial marks for correct approach
- Show all steps for method marks
- Verify your answer mentally
Master All 10 Continuity Types?
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