Continuity — Definition, types of discontinuity
Limits & Continuity
9
JEE Qs
8%
Hard
60
min
Approach continuity problems systematically by first identifying potential points of discontinuity, then rigorously applying the three-condition check (LHL, RHL, f(a)).
🧮 Key Formulas
✅ Key Points for JEE
- 1To check continuity at a point x=a, always evaluate the Left Hand Limit (LHL), Right Hand Limit (RHL), and the function value f(a). All three must exist and be equal.
- 2Functions like polynomials, exponential functions, logarithmic functions (within their domain), and trigonometric functions (within their domain) are continuous everywhere in their respective domains.
- 3For piecewise functions, critical points to check for discontinuity are where the function definition changes, and points where individual function parts might be undefined (e.g., denominator zero).
- 4Understanding the types of discontinuity (removable, non-removable/jump, non-removable/infinite, oscillatory) is crucial for classification, which is often asked in JEE.
- 5Algebra of continuous functions: The sum, difference, product, and quotient (if the denominator is non-zero) of two continuous functions are also continuous. The composition of continuous functions is also continuous.
⚠️ Common Mistakes
- ✕Forgetting to check the function value f(a) along with the limits, or assuming f(a) exists and equals the limit without explicit calculation.
- ✕Incorrectly calculating one-sided limits, especially for functions involving modulus, greatest integer function, or for piecewise definitions.
- ✕Misclassifying the type of discontinuity, for example, confusing a jump discontinuity with a removable discontinuity.
- ✕Not considering the domain of the function while discussing continuity over an interval, leading to incorrect conclusions about points where the function is not defined.
📝 Practice Questions
See allQ7. (2x2−3x+5)(3x−1) 2 limx→∞ is equal to : (3x2+5x+4)√(3x+2)x (1) 2 (2) 2e √3e √3 (3) 2 (4) 2e 3√e 3
Q19.Consider the region R = {(x, y) : x ≤y ≤9 −113 x2, x ≥0}. The area, of the largest rectangle of sides parallel to the coordinate axes and inscribed in R , is: (1) 730 (2) 625 119 111 (3) 821 (4) 567 123 121
Q11.If limx→∞(( 1−e ) ( e − 1+x )) = α, then the value of 1+loge α equals : (1) e−1 (2) e2 (3) e−2 (4) e
Q7. x2 {sin (k1 + 1)x + sin (k2 −1)x}, x < 0 ⎧ If the function f(x) = 4, x = 0 is continuous at x = 0, then k21 + k22 is ⎨ 2 2+k1x x > 0 x loge ( 2+k2x ), ⎩ equal to (1) 20 (2) 5 (3) 8 (4) 10
Q14. IfI(m, n) = ∫10 xm−1(1 −x)n−1dx, m, (1) I(19, 27) (2) I(9, 1) (3) I(1, 13) (4) I(9, 13)
Q9. Let [x] denote the greatest integer function, and let m and n respectively be the numbers of the points, where the function f(x) = [x] + |x −2|, −2 < x < 3, is not continuous and not differentiable. Then m + n is equal to : (1) 6 (2) 8 (3) 9 (4) 7
NCERT Chapters
- Class 12 Maths Ch 5: Continuity and Differentiability