What is continuity of a function at a point?

A function f(x) is continuous at x = c if f(c) is defined and lim(x?c) f(x) = f(c). Equivalently, the left-hand limit, right-hand limit and function value must be equal.

What is the condition for continuity at a point?

The necessary condition is lim(x?c-) f(x) = lim(x?c+) f(x) = f(c). If any one of these conditions fails, the function is discontinuous at c.

What is the difference between continuity and differentiability?

Continuity means the function has no break at a point, while differentiability means the derivative exists at that point. Every differentiable function is continuous, but every continuous function need not be differentiable.

Is every differentiable function continuous?

Yes. If a function is differentiable at x = c, then it is necessarily continuous at x = c. This is an important theorem in Class 12 Mathematics.

How is continuity checked for a piecewise function?

First check continuity on each individual interval, then examine every point where the definition changes. At a boundary point c, verify that the left-hand limit, right-hand limit and f(c) are equal.

What is the derivative of a function defined parametrically?

If x = f(t) and y = g(t), then the derivative is dy/dx = (dy/dt)/(dx/dt), provided dx/dt is not zero.

What is logarithmic differentiation?

Logarithmic differentiation is a method used to differentiate complicated products, quotients or functions in which the variable occurs in both the base and exponent. Taking logarithms first simplifies the differentiation.

What is the chain rule in differentiation?

If y = f(g(x)), then the chain rule gives dy/dx = f'(g(x))g'(x). It is used to differentiate composite functions.

What are the important inverse trigonometric derivatives?

The standard derivatives include d/dx(sin?¹x) = 1/v(1-x²), d/dx(cos?¹x) = -1/v(1-x²), and d/dx(tan?¹x) = 1/(1+x²).

Why is Continuity and Differentiability important for CBSE and JEE exams?

The chapter develops essential calculus skills used in CBSE Board, JEE Main, JEE Advanced and other entrance examinations, particularly for continuity tests, derivatives, composite functions, implicit differentiation, logarithmic differentiation and parametric differentiation.

What is covered in these Class 12 Continuity and Differentiability MCQs?

These MCQs cover continuity, differentiability, limits, derivatives, standard derivatives, chain rule, product rule, quotient rule, inverse trigonometric functions, logarithmic differentiation, and piecewise functions.

How many MCQs are included in this Continuity and Differentiability practice set?

The practice set contains 50 multiple-choice questions with four options, correct answers, and concise explanations.

Are these MCQs based on NCERT Class 12 Mathematics Chapter 5?

Yes, the questions are designed around the concepts and topics covered in NCERT Class 12 Mathematics Chapter 5, Continuity and Differentiability.

Are these MCQs useful for CBSE Class 12 Board exams?

Yes, these MCQs are useful for revising key concepts and practising objective-type questions relevant to CBSE Class 12 Mathematics examinations.

Do the MCQs include answers and explanations?

Yes, every question includes the correct answer and a concise explanation to help students understand the underlying concept.

Continuity And Differentiability | Mathematics Class 12

12 CBSE Marks

★★★★★ Difficulty

9 Topics

Very High JEE Weight

Topics Covered

9 key topics in this chapter

Continuity at a Point & on an Interval

Algebra of Continuous Functions

Differentiability

Chain Rule

Derivatives of Implicit Functions

Exponential & Logarithmic Function Derivatives

Logarithmic Differentiation

Derivatives in Parametric Form

Second Order Derivative

Study Resources

📖 NCERT Notes ✏️ MCQ Practice 🏆 JEE / CBSE PYQs 📝 Exercise Solutions ✅ True / False Quiz

𝑓 Key Formulae

Essential mathematical expressions for this chapter — understand derivations, not just results.

Continuity at a Point

\[\lim_{x\to a}f(x) = f(a)\]

📌 Left limit = right limit = function value

Chain Rule

\[\tfrac{d}{dx}f(g(x)) = f'(g(x))\cdot g'(x)\]

📌 Differentiate outer, then inner

Exponential Derivative

\[\tfrac{d}{dx}e^{x} = e^{x},\quad \tfrac{d}{dx}a^{x}=a^{x}\ln a\]

📌 Base e is its own derivative

Logarithmic Differentiation

\[\ln y = v\ln u \implies \tfrac{y'}{y} = v'\ln u + \tfrac{v u'}{u}\]

📌 Use for uᵛ-type expressions

Parametric Derivative

\[\tfrac{dy}{dx} = \tfrac{dy/dt}{dx/dt}\]

📌 When x and y are both functions of t

🎯 Exam-Ready Insights

Important points to remember — curated from CBSE Board question patterns.

CBSE checks continuity of piecewise functions at the junction point — always compute LHL, RHL, and f(a) separately and compare.

Logarithmic differentiation is the only method for differentiating functions of the form [f(x)]^g(x) — take log of both sides first.

Differentiability implies continuity, but continuity does NOT imply differentiability — |x| at x = 0 is the classic counterexample.

Second-order derivatives (d²y/dx²) for parametric and implicit functions are a common 5-mark question.

🏆 Competitive Exam Strategy

Targeted tips for JEE Main, JEE Advanced, NEET, BITSAT, and CBSE Boards.

JEE Main

Differentiability at a corner point is tested through LHD ≠ RHD; compute both derivatives from first principles when in doubt.

JEE Advanced

Composite chain-rule problems with three or more nested functions are common — differentiate layer by layer, do not rush.

NEET

Basic exponential and logarithmic derivative rules appear in physics-adjacent problems — keep them automatic.

⚠️ Common Mistakes to Avoid

Checking only one-sided continuity instead of both LHL and RHL.

Forgetting to apply the chain rule when differentiating a composite function.

Applying normal differentiation rules to uᵛ expressions instead of logarithmic differentiation.

💡 Key Takeaways

◆

Continuity requires the limit to exist AND equal the function value at that point.

◆

Every differentiable function is continuous, but the converse is false.

◆

The chain rule is the single most-used differentiation tool in this chapter.

Continuity and Differentiability