Continuity and Differentiability
Class 12 Mathematics · Mathematics Part I · Chapter 5
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History
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Summary
This chapter formalises continuity and differentiability. A function \(f\) is continuous at \(x=c\) if \(\lim_{x\to c}f(x)=f(c)\); it is continuous on an interval if continuous at every point. Differentiability is stronger: \(f'(c)\) exists if the limit defining the derivative exists, and differentiability implies continuity (but not conversely). The chapter develops rules for differentiating composite functions (chain rule), implicit functions, inverse trigonometric functions, exponential and logarithmic functions, and functions given in parametric form. Logarithmic differentiation handles products, quotients and variable exponents. Second-order derivatives \(\dfrac{d^2y}{dx^2}\) are introduced, and the chapter closes with the Mean Value Theorems of Rolle and Lagrange, which connect the derivative to the behaviour of the function on an interval.
Continuity of functionsDifferentiability and the chain ruleDerivatives of inverse trig, exponential and log functionsLogarithmic and parametric differentiationMean Value Theorems
Key terms
Continuity at a point
\(\lim_{x\to c}f(x)=f(c)\).
Differentiability
Existence of \(f'(c)=\lim_{h\to0}\dfrac{f(c+h)-f(c)}{h}\).
Chain rule
\(\dfrac{dy}{dx}=\dfrac{dy}{du}\cdot\dfrac{du}{dx}\) for composite functions.
Implicit differentiation
Differentiating an equation in \(x\) and \(y\) without solving for \(y\).
Logarithmic differentiation
Taking \(\log\) before differentiating, useful for \(y=u^v\).
Mean Value Theorem
There is \(c\) with \(f'(c)=\dfrac{f(b)-f(a)}{b-a}\) (Lagrange).
Extra questions & answers
Try one from this chapter
\(f\) is continuous at \(x=c\) if: