Chapter 5 ยท Introduction

Chapter 5 โ€” Continuity and Differentiability

Can you draw the graph without lifting your pen? Does it have a well-defined slope at every point? This chapter makes both questions precise and then builds the full toolkit for differentiating the functions you will meet in Class 12 and JEE.

Drop a cricket ball from a balcony and film it bouncing. Its height changes continuously: the ball never vanishes from one place and reappears at another, so the heightโ€“time graph can be drawn without lifting the pen. But at the instant of each bounce something abrupt does happen โ€” the velocity flips from downward to upward, and the graph has a sharp corner, where it has no single slope. Compare a taxi meter: the fare stays at โ‚น100, then jumps to โ‚น115 the moment the next kilometre starts. That graph cannot be drawn without lifting the pen at all. Smooth, cornered, broken โ€” in Class 11 you met limits and derivatives; in this chapter those three pictures become the precise ideas of continuity and differentiability, and you learn exactly how they are related.

The second half of the chapter is a toolkit. Real functions are rarely as simple as $x^3$ or $\sin x$: they are built in layers, like $\sin(x^2+1)$ or $e^{-0.2t}\cos 3t$; they are hidden inside equations, like a circle $x^2+y^2=25$ that is not a single function $y=f(x)$; they involve powers such as $x^x$ that no earlier rule covers; or they are described by a third variable, the way ISRO describes a rocket by its position at each time $t$. The chain rule, implicit differentiation, the derivatives of $e^x$, $\log x$ and the inverse trigonometric functions, logarithmic differentiation and parametric differentiation handle all of these โ€” and second-order derivatives measure how a rate of change is itself changing. Chapter 6 then puts the whole toolkit to work on rates, maxima and minima.

Limits (Class 11)โ†’Continuityโ†’Differentiabilityโ†’Chain rule & implicit functionsโ†’eหฃ, log x, log differentiationโ†’Parametric & second derivativesโ†’Applications (Ch 6)
๐ŸŽฏ By the end of this chapter you canโ€ฆ
  • test a function for continuity at a point using the left-hand limit, the right-hand limit and the value, and find the constants that make a piecewise function continuous;
  • use the algebra of continuous functions and composition to decide where a function is continuous, and locate every point of discontinuity;
  • decide whether a function is differentiable at a point from its left-hand and right-hand derivatives, and explain why differentiability implies continuity but not conversely;
  • differentiate composite, implicit and inverse trigonometric functions, exponential and logarithmic functions, and functions given in parametric form;
  • use logarithmic differentiation for products, quotients and powers like $x^x$, and find and use second-order derivatives;
  • answer the Board's 1-, 2-, 3- and 5-mark questions on this chapter and the JEE Main / JEE Advanced problems built on it.
Teaser โ€” Draw It Without Lifting the Pen?

The pen traces the graph from left to right, carrying its tangent line. Watch three special points. At $x=-2$ a single point has been moved off the curve; at $x=0$ the curve has a corner; at $x=2$ it jumps. Where must the pen lift (a discontinuity)? Where does the tangent line suddenly flip (no derivative)? Drag the slider to stop at any point.

$x$โ€”
$f(x)$โ€”
slope of tangentโ€”
continuous here?โ€”
differentiable here?โ€”

Sections in this chapter

5.1

Continuity

Continuity at a point and on an interval; removable discontinuities and jumps; the algebra of continuous functions and composition.

$\lim\limits_{x\to c}f(x)=f(c)$
Open section โ†’5.2

Differentiability, the Chain Rule and Implicit Functions

Left and right derivatives; differentiable โ‡’ continuous; the chain rule; implicit differentiation; derivatives of inverse trigonometric functions.

$\dfrac{d}{dx}f(g(x))=f'(g(x))\,g'(x)$
Open section โ†’5.3

Exponential and Logarithmic Functions; Logarithmic Differentiation

The derivatives of $e^x$, $a^x$ and $\log x$; taking logarithms to differentiate products, quotients and powers like $x^x$.

$\dfrac{d}{dx}e^x=e^x,\ \ \dfrac{d}{dx}\log x=\dfrac1x$
Open section โ†’5.4

Parametric Forms and Second-Order Derivatives

Curves described by a parameter, the slope $\tfrac{dy}{dx}$ from $\tfrac{dy}{dt}$ and $\tfrac{dx}{dt}$, and derivatives of derivatives.

$\dfrac{dy}{dx}=\dfrac{dy/dt}{dx/dt}$
Open section โ†’
โœ… Before you start
This chapter builds directly on Class 11 Mathematics, Ch. 12 โ€” Limits and Derivatives: left-hand and right-hand limits, the standard limits $\lim\limits_{x\to0}\dfrac{\sin x}{x}=1$ and $\lim\limits_{x\to a}\dfrac{x^n-a^n}{x-a}=na^{n-1}$, the derivative from first principles, and the sum, product and quotient rules. You also need the modulus, greatest integer and signum functions (Class 11 Mathematics, Ch. 2 โ€” Relations and Functions, and Section 1.2 of this course), trigonometric identities (Class 11 Mathematics, Ch. 3 โ€” Trigonometric Functions), and the principal value branches of the inverse trigonometric functions (Chapter 2 โ€” Inverse Trigonometric Functions). Quick self-test: can you find $\lim\limits_{x\to2}\dfrac{x^2-4}{x-2}$ and differentiate $\dfrac{x}{1+x^2}$ without looking anything up? (Answers: $4$ and $\dfrac{1-x^2}{(1+x^2)^2}$.)
โ† 4.3 Solving Systems of Linear Equations5.1 Continuity โ†’