Continuity and Differentiability

Complete Practice Guide: Continuity and Differentiability Class 12

Continuity and Differentiability is one of the most important Class 12 Mathematics topics because it connects limits with derivatives and prepares students for applications of differentiation. This practice worksheet is designed for repeated revision: every new test generates a fresh set of parameterized questions, while the complete worksheet can also be shuffled. The official NCERT page supplied for this chapter is the Class I–XII textbook portal. citeturn0view0

1. Continuity at a Point

A function f(x) is continuous at x = a when three conditions are satisfied: f(a) exists, the limit as x approaches a exists, and the limit equals the function value. The central formula is lim(x→a) f(x) = f(a). For a piecewise function, first calculate the left-hand limit, right-hand limit and the actual value at the point. If all three agree, continuity holds.

2. Left-Hand and Right-Hand Limits

Continuity problems often use piecewise definitions. Remember lim(x→a⁻) f(x) = lim(x→a⁺) f(x) = f(a). If the two one-sided limits differ, the two-sided limit does not exist and the function cannot be continuous at that point. These questions are excellent practice for parameter-finding problems.

3. Continuous Functions and Operations

Polynomials are continuous for every real number. Rational functions are continuous wherever their denominator is non-zero. Sums, differences and products of continuous functions remain continuous; quotients are continuous at points where the denominator is not zero. This makes it possible to identify continuity quickly without evaluating a complicated limit every time.

4. Differentiability

Differentiability at x = a means that the derivative exists there. Using first principles, f′(a) = lim(h→0) [f(a+h) − f(a)]/h. A function differentiable at a point is necessarily continuous there. However, continuity alone does not guarantee differentiability. Corner points, cusps and certain sharp changes can be continuous but not differentiable.

5. Continuity versus Differentiability

This distinction is a common exam concept. Differentiability ⇒ Continuity, but Continuity ⇏ Differentiability. For example, the absolute-value function is continuous at zero but has unequal left and right derivatives there. When checking a piecewise function, compare the left derivative and right derivative at the joining point.

6. Standard Derivative Rules

Important rules include d/dx(xⁿ) = n xⁿ⁻¹, the constant rule, sum and difference rules, product rule and quotient rule. For a product, (uv)′ = u′v + uv′. For a quotient, (u/v)′ = (vu′ − uv′)/v². Practice identifying the simplest applicable rule before expanding an expression.

7. Chain Rule

For a composite function y = f(g(x)), the chain rule is dy/dx = f′(g(x)) · g′(x). In practical terms, differentiate the outer function while keeping the inner expression unchanged, then multiply by the derivative of the inner expression. This is especially useful for powers, roots and nested trigonometric expressions.

8. Implicit and Logarithmic Differentiation

When x and y occur together, differentiate both sides with respect to x and remember that d/dx(y) = dy/dx. Logarithmic differentiation is useful when variables occur in both bases and exponents or when a product contains several factors. A standard idea is to take logarithms first, simplify, then differentiate.

9. Higher Order Derivatives

If y′ is differentiated again, the result is the second derivative y″. Repeating the process gives higher order derivatives. These questions test algebraic accuracy as well as derivative rules. Keep notation clear and simplify after each differentiation step.

10. How to Score Better

Start with easy continuity and standard-derivative questions, then move to parameter problems and mixed differentiation. In this worksheet, use Check Answer after attempting each question. A correct response turns green; an incorrect response turns red and unlocks Show Answer. Use the explanation to identify the exact step that caused the error. Reattempt similar questions after reviewing the relevant formula.

11. Quick Formula Revision

• Continuity: lim(x→a) f(x) = f(a)
• First principle: f′(a) = lim(h→0)[f(a+h)−f(a)]/h
• Power rule: d(xⁿ)/dx = n xⁿ⁻¹
• Product: (uv)′ = u′v + uv′
• Quotient: (u/v)′ = (vu′−uv′)/v²
• Chain rule: d[f(g(x))]/dx = f′(g(x))g′(x)
• Differentiability ⇒ Continuity

For best results, practise regularly rather than attempting one very large worksheet only once. The mobile-first layout lets students revise on phones, tablets or computers, while the timer and score features can be used for exam-style practice.