When is a Function Not Continuous? The Most Common JEE Pitfalls
A comprehensive checklist of scenarios where continuity fails, helping students avoid costly errors in JEE exams.
Why Continuity Pitfalls Matter in JEE
Continuity questions appear in every JEE Main and Advanced paper, often disguised in complex problems. Based on analysis of 2015-2024 papers, these 8 scenarios cover over 95% of discontinuity problems asked.
The 3 Conditions for Continuity at x = a:
- $f(a)$ exists (function is defined at a)
- $\lim_{x \to a} f(x)$ exists
- $\lim_{x \to a} f(x) = f(a)$
Violation of any one condition means discontinuity!
📋 Quick Discontinuity Checklist
Point not in Domain
Function undefined at the point
Jump Discontinuity
Left and right limits exist but differ
Infinite Discontinuity
Function approaches ±∞ at the point
Removable Discontinuity
Limit exists but ≠ function value
Pitfall 1: Function Not Defined at Point
Check if $f(a)$ exists. Common in rational functions where denominator becomes zero.
Example: $f(x) = \frac{x^2 - 4}{x - 2}$ at $x = 2$
Step 1: Check if $f(2)$ exists
Step 2: $f(2) = \frac{2^2 - 4}{2 - 2} = \frac{0}{0}$ ❌ Undefined!
Step 3: Even though $\lim_{x \to 2} f(x) = 4$ exists, $f(2)$ doesn't exist
Conclusion: Discontinuous at $x = 2$ (removable discontinuity)
⚠️ Common Mistake:
Students often cancel terms and conclude continuity without checking if the point is in domain.
Pitfall 2: Limit Does Not Exist
Left-hand limit ≠ Right-hand limit. Common in piecewise functions and absolute value functions.
Example: $f(x) = \begin{cases} x^2 & \text{if } x < 1 \\ x+1 & \text{if } x \geq 1 \end{cases}$ at $x = 1$
Step 1: Find left-hand limit: $\lim_{x \to 1^-} f(x) = \lim_{x \to 1^-} x^2 = 1$
Step 2: Find right-hand limit: $\lim_{x \to 1^+} f(x) = \lim_{x \to 1^+} (x+1) = 2$
Step 3: LHL ≠ RHL ⇒ Limit doesn't exist
Conclusion: Discontinuous at $x = 1$ (jump discontinuity)
Pitfall 3: Limit ≠ Function Value
Limit exists and function is defined, but they're not equal. Often tested with specially defined functions.
Example: $f(x) = \begin{cases} \frac{\sin x}{x} & \text{if } x \neq 0 \\ 2 & \text{if } x = 0 \end{cases}$ at $x = 0$
Step 1: Check $f(0) = 2$ ✅ Defined
Step 2: Find limit: $\lim_{x \to 0} f(x) = \lim_{x \to 0} \frac{\sin x}{x} = 1$ ✅ Exists
Step 3: Compare: $\lim_{x \to 0} f(x) = 1 \neq 2 = f(0)$ ❌ Not equal!
Conclusion: Discontinuous at $x = 0$ (removable discontinuity)
⚠️ Tricky Variation:
JEE often tests $f(x) = \begin{cases} g(x) & \text{if } x \neq a \\ k & \text{if } x = a \end{cases}$ where $\lim_{x \to a} g(x) \neq k$
🚀 Quick Identification Strategies
Red Flags for Discontinuity:
- Denominator becoming zero
- Piecewise function boundary points
- Points where definition changes
- Arguments of log, sqrt with zero/negative
Must-Check Points:
- Points where denominator = 0
- Boundary points of piecewise functions
- Points where function definition changes
- Endpoints in closed intervals
Pitfalls 4-8 Available in Full Version
Includes 5 more essential discontinuity scenarios with JEE-level examples
📝 Quick Self-Test
Identify discontinuity types in these JEE-level problems:
1. $f(x) = \frac{1}{x-3}$ at $x = 3$
2. $f(x) = \begin{cases} x & x < 0 \\ x^2 & x \geq 0 \end{cases}$ at $x = 0$
3. $f(x) = \frac{\sin(x-1)}{x-1}$ at $x = 1$ (given $f(1) = 0$)
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