Continuity and Differentiability revision notes

• Polynomial & Constant Functions: Always continuous everywhere in $\mathbb{R}$,,.
• Identity Function ($f(x) = x$): Continuous at every real number.
• Rational Functions ($f(x) = p(x)/q(x)$): Continuous everywhere in their domain (i.e., at all points where $q(x) \neq 0$).
• Trigonometric Functions: Sine and Cosine are continuous everywhere. Tangent, cotangent, secant, and cosecant are continuous everywhere in their respective domains,,.
• Modulus Function ($|x|$): Continuous at all real numbers, including at $x = 0$ where the graph turns sharply,.
• Greatest Integer Function ($[x]$): Discontinuous at every integral point. The limit approaching an integer $c$ from the left is $c-1$, and from the right is $c$,. JEE Tip Whenever a piecewise function or GIF is present, check continuity and limits exactly at the integral points or interval boundaries, as these are primary MCQ traps.

If $f$ and $g$ are continuous at $x = c$:

1. $f + g$ and $f - g$ are continuous at $x = c$.
2. $f \cdot g$ is continuous at $x = c$.
3. $f / g$ is continuous at $x = c$ (provided $g(c) \neq 0$).
4. $\lambda f$ is continuous for any real constant $\lambda$.

• Composite Functions: If $g$ is continuous at $c$ and $f$ is continuous at $g(c)$, then the composite function $(f \circ g)(x) = f(g(x))$ is continuous at $c$. JEE Tip In JEE questions involving $f(|x|)$ or $|f(x)|$, use composite function theorems. Since $|x|$ is continuous, if $f(x)$ is continuous, then $f(|x|)$ and $|f(x)|$ are guaranteed continuous.

• Theorem: Every differentiable function is continuous.
• Converse: Every continuous function is NOT necessarily differentiable. (e.g., $f(x) = |x|$ is continuous at $x=0$ but not differentiable there, as LHD = $-1$ and RHD = $1$). JEE Tip Sharp corners or cusps on a continuous graph always indicate points of non-differentiability.

• Exponential Function ($f(x) = b^x$, where $b > 1$): Domain is $\mathbb{R}$, Range is $\mathbb{R}^+$. The point $(0, 1)$ is always on the graph. The graph is strictly increasing and asymptotic to the negative x-axis,.
• Natural Exponential Function ($f(x) = e^x$): Base is $e$ (a number between 2 and 3 derived from the series $1 + 1/1! + 1/2! + ...$).

1. Chain Rule (Composite Functions): If $f = v \circ u$, $t = u(x)$, and both $dt/dx$ and $dv/dt$ exist, then $\frac{df}{dx} = \frac{dv}{dt} \cdot \frac{dt}{dx}$. Can be extended to any number of nested functions (e.g., $\frac{df}{dx} = \frac{dw}{ds} \cdot \frac{ds}{dt} \cdot \frac{dt}{dx}$).
2. Implicit Differentiation: Differentiate both sides of the equation with respect to $x$, applying the chain rule to terms involving $y$ (yielding a $dy/dx$ factor), then group and solve algebraically for $dy/dx$,.
3. Logarithmic Differentiation: Used for functions of the form $y = [u(x)]^{v(x)}$.
   • Take the natural logarithm of both sides: $\log y = v(x) \log[u(x)]$.
   • Differentiate implicitly: $\frac{1}{y} \frac{dy}{dx} = v'(x) \log[u(x)] + v(x) \frac{u'(x)}{u(x)}$.
   • Condition: Both $f(x)$ and $u(x)$ MUST be strictly positive for their logarithms to be defined.
4. Parametric Differentiation: For $x = f(t), y = g(t)$, the derivative is: $\frac{dy}{dx} = \frac{dy/dt}{dx/dt} = \frac{g'(t)}{f'(t)}$, provided $f'(t) \neq 0$.
5. Higher Order Derivatives: Second-order derivative is $\frac{d^2y}{dx^2} = \frac{d}{dx}(\frac{dy}{dx})$, denoted as $f''(x), y'', y_2, \text{ or } D^2y$.

• Rolle's Theorem: If $f(x)$ is continuous on $[a, b]$, differentiable on $(a, b)$, and $f(a) = f(b)$, then there exists at least one $c \in (a, b)$ such that $f'(c) = 0$.
• Lagrange's Mean Value Theorem (LMVT): If $f(x)$ is continuous on $[a, b]$ and differentiable on $(a, b)$, there exists at least one $c \in (a, b)$ such that $f'(c) = \frac{f(b) - f(a)}{b - a}$. JEE Tip LMVT is heavily tested in inequality proofs and finding the number of roots of equations.
• L'Hôpital's Rule: Used for $0/0$ or $\infty/\infty$ indeterminate forms. $\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}$.
• Differentiation of Inverse Functions: If $g(x)$ is the inverse of $f(x)$ (i.e., $f(g(x)) = x$), then $g'(x) = \frac{1}{f'(g(x))}$.
• Differentiation of Determinants: To differentiate a determinant $\Delta(x)$, differentiate row-by-row (or column-by-column), adding the resulting determinants together.

Algebra of Limits/Derivatives:

• Product Rule (Leibnitz): $(uv)' = u'v + uv'$.
• Quotient Rule: $(\frac{u}{v})' = \frac{u'v - uv'}{v^2}$ (where $v \neq 0$).

Standard Derivative Formulae:

• Quotient Rule / Rational Functions: Functions of the form $p(x)/q(x)$ are strictly undefined, and their derivatives are invalid, wherever the denominator $q(x) = 0$,.
• Logarithmic Differentiation Constraints: When transforming $y = [u(x)]^{v(x)}$ into $\log y = v(x)\log u(x)$, $u(x)$ MUST strictly be $>0$. If the base can be negative, standard logarithmic differentiation fails.
• Inverse Trigonometric Derivatives: The derivative of $\sin^{-1}x$ is $\frac{1}{\sqrt{1-x^2}}$. This is undefined at $x = \pm 1$ because the tangent to the inverse sine curve becomes perfectly vertical at the boundaries.

• $f(x) = 1/x$: Approaches $+\infty$ as $x \to 0^+$, approaches $-\infty$ as $x \to 0^-$. Graph consists of two disjoint hyperbolas in Q1 and Q3,,.
• Greatest Integer Function $f(x) = [x]$: A "step" graph. Flat on intervals $[n, n+1)$, jumping up at every integer. Discontinuous and non-differentiable at every integer,.
• Exponential and Logarithmic Graphs: $y = e^x$ passes through $(0,1)$ and stays strictly in Q1 and Q2. $y = \ln x$ passes through $(1,0)$ and stays strictly in Q1 and Q4. They are symmetrical mirror images of each other reflected across the line $y = x$,,.
• Polynomial Growth Comparison: Higher degree polynomials lean closer to the y-axis (grow faster). However, an exponential function (like $10^x$ or $e^x$) will eventually grow faster and overtake ANY polynomial function $x^n$ for large enough $x$,,.

• Infinity Trap: Assuming $+\infty$ or $-\infty$ is a valid limit value. Infinity is NOT a real number. If a limit evaluates to infinity, the limit does not exist. Similarly, continuity implies a finite limit,.
• Drawing Without Lifting Pen Trap: Thinking that "drawing a graph without lifting a pen" is a perfectly rigorous definition of continuity. It is informal. The mathematical rigor strictly requires limit calculation,.

1. Differentiability of $|f(x)|$: Often tests identifying where a function inside a modulus changes sign.
2. Functional Equations: e.g., $f(x+y) = f(x)f(y)$. Students must use the first principle limit definition to establish a differential equation (like $f'(x) = k \cdot f(x)$).
3. Parametric higher-order differentiation: Extremely common in JEE Advanced coordinate geometry integrations.
4. Implicit differentiation coupled with Inverse Trig: Using trigonometric substitutions (e.g., $x = \tan \theta$) to simplify inverse functions before differentiating,.