Continuity and Differentiability revision notes for JEE

Key Concepts & Definitions

Continuity (Informal):: A function is continuous at a fixed point if its graph can be drawn around that point without lifting the pen from the plane of the paper.
Continuity at a Point:: A function fff is continuous at x=cx = cx=c in its domain if the limit of the function at x=cx = cx=c equals the value of the function at x=cx = cx=c. Formula: lim⁡x→c−f(x)=lim⁡x→c+f(x)=f(c)\lim_{x \to c^-} f(x) = \lim_{x \to c^+} f(x) = f(c)limx→c−f(x)=limx→c+f(x)=f(c).
Continuity in an Interval:: A function is continuous if it is continuous at every point in its domain. For a closed interval [a,b][a, b][a,b], fff must be right-continuous at x=ax = ax=a (lim⁡x→a+f(x)=f(a)\lim_{x \to a^+} f(x) = f(a)limx→a+f(x)=f(a)) and left-continuous at x=bx = bx=b (lim⁡x→b−f(x)=f(b)\lim_{x \to b^-} f(x) = f(b)limx→b−f(x)=f(b)),.
Infinity as a Limit:: If a limit shoots up to +∞+\infty+∞ or drops to −∞-\infty−∞, the limit does not exist as a real number, because ±∞\pm\infty±∞ are NOT real numbers,.
Differentiability at a Point:: A function fff is differentiable at ccc if the left-hand derivative (LHD) and right-hand derivative (RHD) are finite and equal. First Principle of Derivative: f′(c)=lim⁡h→0f(c+h)−f(c)hf'(c) = \lim_{h \to 0} \frac{f(c+h) - f(c)}{h}f′(c)=limh→0hf(c+h)−f(c). LHD: lim⁡h→0−f(c+h)−f(c)h\lim_{h \to 0^-} \frac{f(c+h) - f(c)}{h}limh→0−hf(c+h)−f(c) and RHD: lim⁡h→0+f(c+h)−f(c)h\lim_{h \to 0^+} \frac{f(c+h) - f(c)}{h}limh→0+hf(c+h)−f(c).
Explicit vs Implicit Functions:: If a relationship between xxx and yyy can be easily solved to write y=f(x)y = f(x)y=f(x), yyy is an explicit function. If it cannot be easily isolated, yyy is given implicitly,.
Parameter:: When the relation between two variables xxx and yyy is established via a third variable ttt, ttt is called the parameter, yielding parametric equations x=f(t),y=g(t)x = f(t), y = g(t)x=f(t),y=g(t),.

Continuity of Standard Functions

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 TIPWhenever 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.

Algebra of Continuous & Composite Functions

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

$f + g$ and $f - g$ are continuous at $x = c$ .
$f \cdot g$ is continuous at $x = c$ .
$f / g$ is continuous at $x = c$ (provided $g(c) \neq 0$ ).
$\lambda f$ 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 TIPIn 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.

Differentiability and its Relation to Continuity

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 TIPSharp corners or cusps on a continuous graph always indicate points of non-differentiability.

Exponential and Logarithmic Functions

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 $1 + 1/1! + 1/2! + ...$ ).
Logarithmic Function ( $\log_b x$ ): The inverse of the exponential function. If $b^y = x$ $b^{y} = x$ , then $\log_b x = y$ $lo g_{b} x = y$ .
- Domain: Positive real numbers ( $\mathbb{R}^+$ ).
- Range: All real numbers ( $\mathbb{R}$ ).
- The point $(1, 0)$ is always on the graph.
- The graphs of $e^x$ and $\ln x$ are mirror images across the line $y = x$ .
Properties of Logarithms:
1. Base Change: $\log_a p = \frac{\log_b p}{\log_b a}$ .
2. Product Rule: $\log_b (pq) = \log_b p + \log_b q$ .
3. Quotient Rule: $\log_b (p/q) = \log_b p - \log_b q$ .
4. Power Rule: $\log_b (p^n) = n \log_b p$ .
5. Exponential Identity: $x = e^{\log x}$ is valid ONLY for $x > 0$ .JEE TIPAlways verify $x>0$ before applying logarithmic transformations. Applying logs to negative numbers or zero is a lethal trap.

Methods of Differentiation

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., $\frac{df}{dx} = \frac{dw}{ds} \cdot \frac{ds}{dt} \cdot \frac{dt}{dx}$ ).
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 $dy/dx$ ,.
Logarithmic Differentiation: Used for functions of the form $y = [u(x)]^{v(x)}$ $y = [u (x)]^{v (x)}$ .
- 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.
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 $f'(t) \neq 0$ .
Higher Order Derivatives: Second-order derivative is $\frac{d^2y}{dx^2} = \frac{d}{dx}(\frac{dy}{dx})$ , denoted as $f''(x), y'', y_2, \text{ or } D^2y$ .

Mean Value Theorems & Advanced JEE Concepts

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 TIPLMVT 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.

Formulae, Equations & Units

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:

Function $f(x)$	Derivative $f'(x)$	Conditions / Domain	Source
$x^n$	$n x^{n-1}$	$x \in \mathbb{R}$ for integer $n>0$
$\sin x$	$\cos x$	$x \in \mathbb{R}$
$\cos x$	$-\sin x$	$x \in \mathbb{R}$
$\tan x$	$\sec^2 x$	$x \neq (2n+1)\frac{\pi}{2}$
$e^x$	$e^x$	$x \in \mathbb{R}$
$\log x$	$1/x$	$x > 0$
$a^x$	$a^x \log a$	$a > 0$
$\sin^{-1} x$	$\frac{1}{\sqrt{1-x^2}}$	$x \in (-1, 1)$	,
$\cos^{-1} x$	$\frac{-1}{\sqrt{1-x^2}}$	$x \in (-1, 1)$
$\tan^{-1} x$	$\frac{1}{1+x^2}$	$x \in \mathbb{R}$

Conditions & Limitations

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.

Important Graphs & Diagrams

$f(x) = 1/x$ : 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 $x$ ,,.

⚠️ COMMON MISCONCEPTIONS & SIGN CONVENTIONS

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,.
Parametric Double Derivative Error: A massive trap is assuming that $\frac{d^2y}{dx^2} = \frac{d^2y/dt^2}{d^2x/dt^2}$ $\frac{d ^{2} y}{d x ^{2}} = \frac{d ^{2} y / d t ^{2}}{d ^{2} x / d t ^{2}}$ . This is fundamentally false.
- Correct rule: $\frac{d^2y}{dx^2} = \frac{d}{dt}\left(\frac{dy/dt}{dx/dt}\right) \cdot \frac{dt}{dx}$ .

Previous Year JEE Topics

Differentiability of $|f(x)|$ : Often tests identifying where a function inside a modulus changes sign.
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 $f'(x) = k \cdot f(x)$ ).
Parametric higher-order differentiation: Extremely common in JEE Advanced coordinate geometry integrations.
Implicit differentiation coupled with Inverse Trig: Using trigonometric substitutions (e.g., $x = \tan \theta$ ) to simplify inverse functions before differentiating,.

Top 10 JEE MCQ Traps

Misconception $\rightarrow$ $\to$ Differentiability implies continuity, so continuity implies differentiability.
- Correct Understanding $\rightarrow$ Continuity is a necessary but not sufficient condition for differentiability. Check sharp corners ( $|x|$ at $0$ where LHD $\neq$ RHD).
Misconception $\rightarrow$ $\to$ For $y = x^x$ $y = x^{x}$ , the derivative is $x \cdot x^{x-1}$ $x \cdot x^{x - 1}$ (power rule) or $x^x \log x$ $x^{x} lo g x$ (exponential rule).
- Correct Understanding $\rightarrow$ Both rules are invalid because neither the base nor the exponent is constant. You MUST use logarithmic differentiation or rewrite as $y = e^{x \ln x}$ .JEE TIPNever apply polynomial/exponential rules to variable^variable. Always take logs.
Misconception $\rightarrow$ $\to$ $\frac{d^2y}{dx^2}$ $\frac{d ^{2} y}{d x ^{2}}$ of parametric equations $x(t), y(t)$ $x (t), y (t)$ is simply $\frac{y''(t)}{x''(t)}$ $\frac{y ^{''} ( t )}{x ^{''} ( t )}$ .
- Correct Understanding $\rightarrow$ $\frac{d^2y}{dx^2} = \frac{d}{dx}(\frac{dy}{dx}) = \frac{d}{dt}(\frac{y'(t)}{x'(t)}) \times \frac{dt}{dx}$ . Don't forget the $dt/dx$ chain rule factor at the end!
Misconception $\rightarrow$ $\to$ $\log_b x$ $lo g_{b} x$ domain is just $x \neq 0$ $x \neq = 0$ .
- Correct Understanding $\rightarrow$ The domain is strictly $x > 0$ . Negative logs are undefined in reals. Taking the log of an implicitly negative variable will wreck domain bounds.
Misconception $\rightarrow$ $\to$ $y = \sin^{-1}(\sin x) \implies y' = 1$ $y = sin^{- 1} (sin x) ⟹ y^{'} = 1$ always.
- Correct Understanding $\rightarrow$ $\sin^{-1}(\sin x) = x$ ONLY in the principal domain $[-\pi/2, \pi/2]$ . Outside this, the graph is a zigzag line and the derivative alternates between $1$ and $-1$ depending on the interval.
Misconception $\rightarrow$ $\to$ $\lim_{x\to 0} 1/x = \infty$ $lim_{x \to 0} 1/ x = \infty$ , so the limit exists.
- Correct Understanding $\rightarrow$ $\infty$ is not a real number. Also, RHL is $+\infty$ and LHL is $-\infty$ . The limit does not exist mathematically,.
Misconception $\rightarrow$ $\to$ $f(x)g(x)$ $f (x) g (x)$ is discontinuous if $f(x)$ $f (x)$ or $g(x)$ $g (x)$ is discontinuous.
- Correct Understanding $\rightarrow$ If $f(x)$ is continuous and $g(x)$ is discontinuous at $x=c$ , their product CAN be continuous if $f(c) = 0$ . E.g., $x \cdot \sin(1/x)$ at $x=0$ .
Misconception $\rightarrow$ $\to$ $x = e^{\ln x}$ $x = e^{l n x}$ applies for all real numbers.
- Correct Understanding $\rightarrow$ It only holds true for $x > 0$ . For $x \le 0$ , the right-hand side is undefined.
Misconception $\rightarrow$ $\to$ Differentiating $\sqrt{x^2}$ $x^{2}$ simply gives $1$ $1$ .
- Correct Understanding $\rightarrow$ $\sqrt{x^2} = |x|$ , so the derivative is $+1$ for $x>0$ and $-1$ for $x<0$ . It is not differentiable at $x=0$ .
Misconception $\rightarrow$ $\to$ The sum of two non-differentiable functions is always non-differentiable.
- Correct Understanding $\rightarrow$ The sum can be perfectly differentiable. E.g., $f(x) = |x|$ and $g(x) = -|x|$ are non-differentiable at $0$ , but their sum $h(x) = 0$ is differentiable everywhere.