Continuity and Differentiability Class 12: notes and important questions

Continuity and Differentiability — revision notes

1. Continuity

\(f\) is continuous at \(x = c\) (a point of its domain) if \(\displaystyle\lim_{x\to c^-}f(x) = \lim_{x\to c^+}f(x) = f(c)\). A function is continuous if it is continuous at every point of its domain; points outside the domain are not called discontinuities.

• Polynomials, \(\sin x, \cos x\), \(e^x\), \(\log x\) \((x>0)\), \(|x|\), rational functions (where defined) are continuous.
• Sums, differences, products, quotients (denominator \(\ne 0\)) and composites of continuous functions are continuous.
• Useful limits: \(\displaystyle\lim_{x\to0}\frac{\sin x}{x} = 1\), \(\displaystyle\lim_{x\to0}\frac{1-\cos x}{x^2} = \frac12\).
• \([x]\) (greatest integer) is discontinuous exactly at the integers.

2. Differentiability

\(f'(c) = \displaystyle\lim_{h\to0}\frac{f(c+h) - f(c)}{h}\) must exist, i.e. LHD \(=\) RHD (finite). Differentiable \(\Rightarrow\) continuous; the converse is false (e.g. \(|x|\) at \(0\)). For a piecewise function, first check continuity, then compare the one-sided derivatives.

3. Rules

• Chain rule: \(\dfrac{d}{dx}f(g(x)) = f'(g(x))\,g'(x)\).
• \(\dfrac{d}{dx}\sin^{-1}x = \dfrac{1}{\sqrt{1-x^2}}\), \(\dfrac{d}{dx}\cos^{-1}x = -\dfrac{1}{\sqrt{1-x^2}}\) \((|x| \lt 1)\), \(\dfrac{d}{dx}\tan^{-1}x = \dfrac{1}{1+x^2}\).
• \(\dfrac{d}{dx}e^x = e^x\), \(\dfrac{d}{dx}a^x = a^x\log a\), \(\dfrac{d}{dx}\log|x| = \dfrac1x\).
• Implicit: differentiate every term w.r.t. \(x\), treating \(y\) as a function of \(x\) (\(\frac{d}{dx}y^2 = 2y\frac{dy}{dx}\)).
• Logarithmic differentiation for \([u(x)]^{v(x)}\): \(\log y = v\log u\). For a sum of such terms, differentiate each term separately.
• Parametric: \(\dfrac{dy}{dx} = \dfrac{dy/dt}{dx/dt}\); \(\dfrac{d^2y}{dx^2} = \dfrac{\frac{d}{dt}\left(\frac{dy}{dx}\right)}{dx/dt}\) (not \(\frac{d^2y/dt^2}{d^2x/dt^2}\)).

4. Inverse-trig simplification

Substitute \(x = \sin\theta, \cos\theta\) or \(\tan\theta\), simplify, and check that the resulting angle lies in the principal range before writing \(\sin^{-1}(\sin\alpha) = \alpha\). If it does not, adjust (e.g. \(\pi - \alpha\)); the derivative can change sign across such points.

Worked example 1

Is \(f(x) = \begin{cases} 3x - 2, & x \le 2 \\ x^2, & x > 2\end{cases}\) differentiable at \(2\)? \(f(2) = 4\), right limit \(4\): continuous. LHD \(= 3\), RHD \(= 4\): not differentiable.

Worked example 2

\(y = (\cos x)^{\sin x}\), \(0 \lt x \lt \frac\pi2\): \(\log y = \sin x\log\cos x \Rightarrow \dfrac{dy}{dx} = (\cos x)^{\sin x}\left[\cos x\log\cos x - \dfrac{\sin^2 x}{\cos x}\right]\).

Worked example 3

\(x = at^2,\ y = at^3\): \(\dfrac{dy}{dx} = \dfrac{3at^2}{2at} = \dfrac{3t}{2}\); \(\dfrac{d^2y}{dx^2} = \dfrac{3/2}{2at} = \dfrac{3}{4at}\).

Common errors

• Declaring a function differentiable because the two formulas "join up" — continuity is not enough.
• Using \(\log(u+v) = \log u + \log v\) (false). Take logs only of products, quotients and powers.
• Treating \(x^x\) like \(x^n\) or \(a^x\): neither rule applies; use logarithms.
• Forgetting \(\frac{dy}{dx}\) on \(y\)-terms in implicit differentiation, or the inner derivative in the chain rule.
• Dividing \(\frac{d^2y}{dt^2}\) by \(\frac{d^2x}{dt^2}\) for a parametric second derivative.

Board tips

• In "find \(k\)" continuity questions, write LHL, RHL and \(f(c)\) explicitly — each carries marks.
• In "prove that" second-order questions, differentiate the simplest form (e.g. multiply out \(y_1\sqrt{1-x^2} = \ldots\)) before differentiating again.
• State the interval restriction whenever you use an inverse-trig substitution.
• Rolle’s theorem and the Mean Value Theorem are no longer in the syllabus — do not spend time on them.