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Continuity and Differentiability
This chapter connects limits with the behaviour of functions near a point and then builds the derivative as a precise rate of change. Continuity checks whether the function value agrees with the limiting value, while differentiability checks whether the left-hand and right-hand slopes agree. A major exam skill in this chapter is knowing which rule applies under which condition. For example, differentiability implies continuity, but continuity alone does not imply differentiability. Similarly, chain rule, implicit differentiation, logarithmic differentiation, and parametric differentiation each have a specific structure to identify before starting calculation. The chapter is formula-rich, but marks are usually lost in conditions and algebra. Students must state one-sided limits, left and right derivatives, domains of inverse trigonometric and logarithmic functions, and non-zero denominator conditions wherever needed. Most long-answer questions combine two or more methods, such as continuity with differentiability, chain rule with inverse trigonometric functions, or logarithmic differentiation with product and quotient forms.
Difficulty
Medium
Study time
70-90 min
Plan by time
Pick the window that matches what you have right now.
If you have 15 min
Last-pass revision
Skim the Quick Revision table — definitions, formulas, and the traps board examiners reuse.
Open Quick RevisionIf you have 45 min
Targeted practice
Read the high-priority concepts, then take the chapter MCQ quiz to find weak spots.
Start MCQ QuizIf you have 70 min
First full pass
Walk every concept in chapter order, then revise and quiz. Best for the first time you study this chapter.
Open Key ConceptsChapter Learning Map
Start with one of the buckets below, then open the full map when you want the complete concept roadmap.
Key Concepts
Concepts grouped the way the chapter is taught — open the bucket that matches what you want to revise.
Core Concepts
high priorityOpen the chapter concepts in a clean revision order.
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).
Differentiability and Continuity
A function f is differentiable at x = a if its left-hand derivative and right-hand derivative at a both exist and are equal.
Chain Rule for Composite Functions
The chain rule gives the derivative of a composite function f(g(x)) by differentiating the outer function at g(x) and multiplying by the derivative of g(x).
Implicit Differentiation
Implicit differentiation is used when x and y are related by an equation and y is not first written explicitly as a function of x.
Derivatives of Inverse Trigonometric, Exponential, and Logarithmic Functions
This concept covers standard derivatives of inverse trigonometric functions, exponential functions, and logarithmic functions, along with their domain conditions.
Logarithmic Differentiation
Logarithmic differentiation is a method where logarithms are taken on both sides before differentiating, especially for variable powers and complicated products or quotients.
Parametric and Second-Order Derivatives
For parametric equations x = f(t), y = g(t), the derivative dy/dx is found as (dy/dt)/(dx/dt), provided dx/dt is not zero. The second-order derivative measures the rate of change of dy/dx with respect to x.
Exam Intelligence
Use this section to decide what deserves the most revision time.
High Probability Topics
- Continuity at a Point
- Differentiability and Continuity
- Chain Rule for Composite Functions
- Implicit Differentiation
- Derivatives of Inverse Trigonometric, Exponential, and Logarithmic Functions
- Logarithmic Differentiation
- Parametric and Second-Order Derivatives
Common Traps
- Assuming continuity proves differentiability.
- Using the wrong branch to calculate f(a) in a piecewise function.
- Forgetting the inner derivative in chain rule.
- Differentiating y^n as ny^(n-1) instead of ny^(n-1)dy/dx during implicit differentiation.
- Missing the negative sign in the derivative of cos^-1 x.
- Forgetting to multiply by y at the end of logarithmic differentiation.
- Calling d/dt(dy/dx) the second derivative without dividing by dx/dt.
Likely Question Types
- MCQ: concept checks, applications, and common mistakes
- Very short answer: definitions, formulas, conditions, or terms
- Short answer: process, diagram, reasoning, or worked method
- Case-based: chapter scenario with linked subparts
Quick Revision
Concept, formula or equation to remember, and the trap that loses marks — in one scannable view.
- Continuity requires the left limit, right limit, and function value to match.
- Differentiability is stronger than continuity and requires equal one-sided derivatives.
- Most derivative calculations in this chapter depend on chain rule either directly or indirectly.
- Implicit and logarithmic differentiation are methods chosen by expression structure, not by question length.
- Parametric derivatives require careful conversion from t-rates to x-rates.
- The most reliable way to protect marks is to state conditions before division, logarithms, and one-sided comparisons.
- 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).
- Differentiability and Continuity: A function f is differentiable at x = a if its left-hand derivative and right-hand derivative at a both exist and are equal.
Practice
Use short concept checks first, then move into the full chapter test.
Free Chapter MCQ Quiz
Try a 15-question quiz from this chapter. Get instant score and unlock concept-wise analytics.
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