Which condition must be met for a function f(x) to be continuous at x0?

The limit as x approaches x0 equals f(x0).

Is it true that An infinitesimal is just a very small number.?

An infinitesimal is a function or sequence whose values approach zero during a specified limiting process, not a fixed small number.

Is it true that If a function is continuous on an interval, it must attain its maximum and minimum values.?

This is only true for continuous functions on closed and bounded intervals (Extreme Value Theorem); open intervals or discontinuities may prevent attainment.

Is it true that If a function is differentiable, it must be continuous.?

Differentiability at a point implies continuity at that point, but the converse is false; a function can be continuous without being differentiable (e.g., |x| at x=0).

Calculus Fundamentals: Functions, Limits, and Derivatives

Calculus provides the tools to understand change and accumulation, from the fundamental properties of functions and their limits to the precise measurement of rates of change through differentiation.

Core Principles

A function maps each element in its domain to exactly one element in its codomain, with the set of all outputs forming the range.
Limits describe the behavior of a function or sequence as its input approaches a certain value or infinity, forming the basis for continuity and differentiation.
Continuity at a point means a function's value at that point equals its limit at that point, implying no breaks or jumps in the graph.
The derivative measures the instantaneous rate of change of a function, representing the slope of the tangent line to its graph.
Differentiation rules, including sum, product, quotient, chain, and inverse function rules, provide systematic ways to find derivatives of complex functions.
Higher-order derivatives (second, third, etc.) describe the rate of change of the rate of change, revealing information about concavity and inflection points.

Action Steps

1. Understand the definition of a function: ensure each input maps to exactly one output.
2. Analyze limits: determine function behavior as input approaches a value or infinity.
3. Check continuity: verify if $lim_{x\to x_0} f(x) = f(x_0)$ to ensure no breaks in the graph.
4. Calculate derivatives: use limit definition or differentiation rules to find the instantaneous rate of change.
5. Apply differentiation rules: systematically find derivatives of sums, products, quotients, compositions, and inverse functions.
6. Compute higher-order derivatives: differentiate the derivative function to find second, third, and nth derivatives.
7. Use implicit differentiation: find $dy/dx$ when y is not explicitly defined as a function of x.
8. Differentiate parametrically: use $dy/dx = (dy/dt) / (dx/dt)$ for functions defined by parametric equations.
9. Identify points of discontinuity: check for undefined points, non-existent limits, or limits not matching function values.

Formulas

Definition of a function: $f: D_f \rightarrow \mathbb{R}$ is a function if for every $x \in D_f$, there is exactly one $f(x) \in \mathbb{R}$
Range of a function: $R_f = f(D_f) = \{y \in \mathbb{R} | y = f(x), x \in D_f\}$
One-to-one function: $x_1 \neq x_2 \Rightarrow f(x_1) \neq f(x_2)$
Inverse function: $f^{-1}(y) = x \iff f(x) = y$
Composite function: $(f \circ g)(x) = f(g(x))$
Bounded above: $f(x) \leq M, x \in D_f$
Bounded below: $f(x) \geq m, x \in D_f$
Bounded function: $|f(x)| \leq K, x \in D_f$
Strictly increasing: $f(x_1) < f(x_2)$ for $x_1 < x_2$
Strictly decreasing: $f(x_1) > f(x_2)$ for $x_1 < x_2$
Even function: $f(-x) = f(x)$
Odd function: $f(-x) = -f(x)$
Periodic function: $f(x + \ell) = f(x)$ for some non-zero real number $\\ell$
Power function: $f(x) = x^\alpha, \alpha \in \mathbb{R}$
Exponential function: $f(x) = a^x, a > 0, a \neq 1$
Logarithmic function: $f(x) = \log_a x, a > 0, a \neq 1$
Limit of a sequence (ε–N definition): $|x_n - a| < \varepsilon$ whenever $n \geq N$
Limit of a function at a finite point (ε–δ definition): $0 < |x - x_0| < \delta \Rightarrow |f(x) - A| < \varepsilon$
Limit at infinity: $|x| > X \Rightarrow |f(x) - A| < \varepsilon$
Infinite quantity: $0 < |x - x_0| < \delta \Rightarrow |f(x)| > M$
Sandwich theorem for sequences: $y_n \leq x_n \leq z_n$ and $lim y_n = lim z_n = A \Rightarrow lim x_n = A$
Sandwich theorem for functions: $g(x) \leq f(x) \leq h(x)$ and $lim g(x) = lim h(x) = A \Rightarrow lim f(x) = A$
Fundamental trigonometric limit: $lim_{x\to 0} \frac{\sin x}{x} = 1$
Monotone convergence theorem for sequences: Every bounded monotonic sequence has a finite limit.
Euler's number: $e := \lim_{n\to\infty} \left(1 + \frac{1}{n}\right)^n$
Continuity at a point: $lim_{x\to x_0} f(x) = f(x_0)$.
One-sided continuity: $lim_{x\to x_0^-} f(x) = f(x_0)$ or $lim_{x\to x_0^+} f(x) = f(x_0)$
Point of discontinuity: (i) $f(x_0)$ undefined; (ii) $lim_{x\to x_0} f(x)$ does not exist; (iii) $lim_{x\to x_0} f(x)$ exists but $lim_{x\to x_0} f(x) \neq f(x_0)$
Infinite discontinuity: At least one one-sided limit is $±\infty$
Oscillatory discontinuity: Function oscillates as $x \to x_0$ with no finite or infinite limit.
Jump discontinuity: One-sided limits exist, are finite, but unequal.
Removable discontinuity: One-sided limits exist, are equal to $L \in \mathbb{R}$, but $f(x_0) \neq L$ or $f(x_0)$ is undefined.
Algebraic laws of limits: $lim(f \pm g) = lim f \pm lim g$, $lim(fg) = (lim f)(lim g)$, $lim \frac{f}{g} = \frac{lim f}{lim g}$ (if $lim g \neq 0$)
Preservation of order: If $f(x) \geq g(x)$ and $lim f(x) = A, lim g(x) = B$, then $A \geq B$
Equivalent infinitesimals: $\\alpha(x) \sim \beta(x) \iff lim \frac{\\beta(x)}{\\alpha(x)} = 1$
Derivative at a point: $f'(x_0) = lim_{\\Delta x \to 0} \frac{f(x_0 + \Delta x) - f(x_0)}{\\Delta x}$
Derivative function: $f'(x) = lim_{\\Delta x \to 0} \frac{f(x + \Delta x) - f(x)}{\\Delta x}$
Derivative of power function: $\\frac{d}{dx}(x^\alpha) = \alpha x^{\\alpha-1}$
Derivative of trigonometric functions: $\\(sin x)' = cos x$, $(\\cos x)' = -\\sin x$, $(\\tan x)' = \frac{1}{\\cos^2 x}$
Derivative of inverse trigonometric functions: $(\\arcsin x)' = \frac{1}{\\sqrt{1-x^2}}$, $(\\arccos x)' = -\frac{1}{\\sqrt{1-x^2}}$, $(\\arctan x)' = \frac{1}{1+x^2}$
Derivative of exponential and logarithmic functions: $(\\a^x)' = a^x \ln a$, $(e^x)' = e^x$, $(\\log_a x)' = \frac{1}{x \ln a}$, $(\\ln x)' = \frac{1}{x}$
One-sided derivatives: $f'_-(x_0) = lim_{\\Delta x \to 0^-} \frac{f(x_0 + \Delta x) - f(x_0)}{\\Delta x}$, $f'_+(x_0) = lim_{\\Delta x \to 0^+} \frac{f(x_0 + \Delta x) - f(x_0)}{\\Delta x}$
Differentiability implies continuity: If $f(x)$ is differentiable at $x_0$, it is continuous at $x_0$
Tangent line equation: $y - y_0 = f'(x_0)(x - x_0)$
Normal line equation: $y - y_0 = -\frac{1}{f'(x_0)}(x - x_0)$ (if $f'(x_0) \neq 0$)
Algebraic rules of differentiation: $(f \pm g)' = f' \pm g'$, $(fg)' = f'g + fg'$, $(\frac{f}{g})' = \frac{f'g - fg'}{g^2}$
Chain rule: $(\frac{dy}{dx}) = \frac{dy}{du} \frac{du}{dx}$
Second derivative: $y'' = f''(x) = \frac{d^2y}{dx^2}$
nth derivative: $y^{(n)} = f^{(n)}(x) = \frac{d^n y}{dx^n}$
Derivative of implicit function: Differentiate both sides w.r.t. x, solve for $dy/dx$
Logarithmic differentiation: Take $ln$ of both sides, differentiate implicitly.
Derivative of parametric function: $\\frac{dy}{dx} = \frac{dy/dt}{dx/dt}$
Second derivative of parametric function: $\\frac{d^2y}{dx^2} = \frac{d}{dt}(\\frac{dy}{dx}) / \frac{dx}{dt}$

Key Terms

Function: A relation that assigns exactly one output value for each input value from its domain.
Domain: The set of all possible input values for a function.
Range: The set of all possible output values of a function.
Limit: The value a function or sequence approaches as the input or index approaches a certain value or infinity.
Continuity: A function is continuous at a point if its limit at that point exists and equals the function's value at that point.
Derivative: The instantaneous rate of change of a function with respect to its variable, representing the slope of the tangent line.
Infinitesimal: A function or sequence whose limit is zero.
Infinite quantity: A function whose absolute value grows without bound as the input approaches a certain value or infinity.
Equivalent infinitesimals: Two infinitesimals whose ratio approaches 1 as they tend to zero.
Discontinuity: A point where a function is not continuous, categorized as infinite, oscillatory, jump, or removable.

Pro Tips

When dealing with limits, always check if direct substitution yields an indeterminate form (0/0, ∞/∞) before applying other methods.
The ε–N and ε–δ definitions of limits are crucial for rigorous proofs; understand the roles of ε (arbitrary smallness) and N or δ (sufficient size).
Equivalent infinitesimals (e.g., sin x ~ x as x -> 0) can significantly simplify limit calculations by allowing substitution.
For functions with piecewise definitions, analyze limits and continuity separately for each piece and at the boundary points.
The derivative of a function represents its instantaneous rate of change; its sign indicates whether the function is increasing or decreasing.
Differentiability implies continuity, but continuity does not imply differentiability (e.g., |x| at x=0).
When differentiating implicitly, remember to treat y as a function of x and apply the chain rule.
Parametric differentiation $dy/dx = (dy/dt) / (dx/dt)$ is useful when x and y are both functions of a third variable (parameter).

Pitfalls to Avoid

Assuming a function is continuous or differentiable without verifying the conditions (e.g., checking limits, one-sided derivatives).
Incorrectly applying limit laws, especially with indeterminate forms.
Confusing the limit of a function as x approaches 0 with the limit as x approaches infinity (e.g., sin(x)/x).
Errors in algebraic manipulation when simplifying expressions before or during differentiation.
Forgetting to apply the chain rule when differentiating composite functions.
Misinterpreting the conditions for theorems like the Extreme Value Theorem or Bolzano's Theorem (e.g., requiring a closed, bounded interval).

Myth vs Reality

An infinitesimal is just a very small number.: An infinitesimal is a function or sequence whose values approach zero during a specified limiting process, not a fixed small number.
If a function is continuous on an interval, it must attain its maximum and minimum values.: This is only true for continuous functions on closed and bounded intervals (Extreme Value Theorem); open intervals or discontinuities may prevent attainment.
If a function is differentiable, it must be continuous.: Differentiability at a point implies continuity at that point, but the converse is false; a function can be continuous without being differentiable (e.g., |x| at x=0).

Real World Examples

A particle's position is given by s = f(t).: The derivative, f'(t), represents the instantaneous velocity of the particle at time t.
A function f(x) = tan x.: It is continuous on its domain {x | x ≠ kπ + π/2, k ∈ Z}, having infinite discontinuities at points where cos x = 0.
The equation x = a(t - sin t), y = a(1 - cos t) describes a cycloid.: The second derivative $d^2y/dx^2$ can be found using parametric differentiation rules.

Statistics

Euler's number (e) decimal expansion begins: e = 2.718281828459...

People

Peter Gustav Lejeune Dirichlet: German mathematician known for contributions to number theory and mathematical analysis, associated with the Dirichlet function.
Carl Friedrich Gauss: German mathematician known for foundational contributions to number theory and mathematics, credited with Gauss's formula for arithmetic series sums.
Bernard Bolzano: Czech mathematician, philosopher, and theologian who helped establish rigorous foundations of modern mathematical analysis, known for Bolzano's Theorem.

Quiz

What is the derivative of f(x) = x^n for a positive integer n?: nx^(n-1)
Which condition must be met for a function f(x) to be continuous at x0?: The limit as x approaches x0 equals f(x0).
What type of discontinuity occurs if lim x→x0- f(x) = A and lim x→x0+ f(x) = B, with A ≠ B?: Jump discontinuity