Chapter 5: Continuity and Differentiability

A function is continuous at a point if its graph has no break there; it is differentiable if the graph has a well-defined tangent. These two notions are the foundation of calculus. In Class XII we sharpen the intuitive picture into precise epsilon-delta-free definitions using limits, prove the key theorems, and build a complete toolkit for differentiating any expression you are likely to meet in school or competitive examinations.

The chapter rests on the limit machinery from Class XI. You will use one-sided limits constantly: a function is continuous at $a$ iff its limit at $a$ equals its value at $a$, and differentiability adds the requirement that the left and right derivatives agree. From these definitions flow the basic theorems: every differentiable function is continuous (but not conversely), polynomial and rational functions are differentiable on their domains, and the algebraic and chain rules let you differentiate compositions.

For board examinations the focus is computational: differentiate everything from $\sin(\log x)$ to $x^x$. JEE asks deeper questions: where is $f(x) = |x| + |x - 1|$ non-differentiable, prove a function satisfies the Mean Value Theorem on a given interval, deduce monotonicity from sign of $f'$. Both tracks emphasise the same skills, just at different depths.

The chapter is long, but it has a clear structure. First continuity, then differentiability, then the rules of differentiation, then advanced techniques (implicit, parametric, logarithmic), and finally the existence theorems (Rolle, Lagrange MVT). Practise each section before moving on. Above all, build the reflex of checking domains before applying rules , many "errors" in differentiation are actually domain confusions.

Mastery of this chapter is the prerequisite for the next two chapters (applications of derivatives and integrals). The differentiation rules below appear in every problem you will solve in Class XII calculus.

What's inside

1. Continuity , limits, one-sided continuity, types of discontinuity.
2. Differentiability , derivatives from first principles, relation to continuity.
3. Rules of differentiation , sum, product, quotient, chain.
4. Implicit differentiation , when $y$ is defined implicitly.
5. Logarithmic differentiation , for products, quotients, and $f^g$.
6. Parametric and second derivatives , chains and higher orders.
7. Mean value theorems , Rolle, Lagrange and applications.

Key results / Formula card

Rules: $(f \pm g)' = f' \pm g'$, $(fg)' = f'g + fg'$, $(f/g)' = (f'g - fg')/g^2$, $(f \circ g)'(x) = f'(g(x)) g'(x)$.

How to read this chapter

Treat continuity as a domain check, differentiability as a smoothness check. Memorise the table of derivatives, then practise composing them via the chain rule. The last subtopic , Mean Value Theorems , reveals the deep reason why derivatives matter.