On the Foundations of Mathematical Analysis
A Tutorial Introduction with Examples
Department of Mathematics
Abstract
This paper serves as both a tutorial introduction to fundamental concepts in mathematical analysis and a demonstration of the markdown-academic document format. We cover essential topics including limits, continuity, differentiation, and integration, while showcasing features such as theorem environments, cross-references, citations, and mathematical typesetting. The goal is to illustrate how academic papers can be written using simple, readable Markdown syntax while maintaining the rigor and formatting standards expected of mathematical writing.
Keywords: mathematical analysis, calculus, limits, continuity, markdown-academic
1 Introduction
Mathematical analysis forms the rigorous foundation of calculus, providing precise definitions and proofs for concepts that are often introduced intuitively. This paper demonstrates how markdown-academic can be used to write mathematical documents with proper theorem environments, equations, and cross-references.
The development of analysis in the 19th century, particularly through the work of Cauchy[1], Weierstrass[2], and Riemann[3], established the epsilon-delta formalism that we use today.
The structure of this paper is as follows: Section 2 covers limits and continuity, Section 3 discusses differentiation, and Section 4 introduces integration. We conclude in Section 5.
2 Limits and Continuity
2.1 Definition of Limits
The concept of a limit is fundamental to all of analysis. We begin with the formal epsilon-delta definition.
Definition 1 (Limit of a Function)
Let $f: D \to \mathbb{R}$ be a function and let $a$ be a limit point of $D$. We say that the limit of $f(x)$ as $x$ approaches $a$ equals $L$, written
if for every $\varepsilon > 0$ there exists a $\delta > 0$ such that for all $x \in D$:
Example 1
Consider $f(x) = x^2$. We claim that $\lim_{x \to 2} x^2 = 4$.
Verification: Given $\varepsilon > 0$, we need $|x^2 - 4| < \varepsilon$ when $|x - 2| < \delta$. Note that $|x^2 - 4| = |x - 2| \cdot |x + 2|$. If we restrict $|x - 2| < 1$, then $|x + 2| < 5$, so $|x^2 - 4| < 5|x - 2|$. Choosing $\delta = \min(1, \varepsilon/5)$ works.
2.2 Continuity
Continuity captures the intuitive notion that small changes in input produce small changes in output.
Definition 2 (Continuity)
A function $f: D \to \mathbb{R}$ is continuous at a point $a \in D$ if
We say $f$ is continuous on $D$ if it is continuous at every point in $D$.
Theorem 1 (Composition of Continuous Functions)
If $f$ is continuous at $a$ and $g$ is continuous at $f(a)$, then $g \circ f$ is continuous at $a$.
Proof.
Let $\varepsilon > 0$ be given. Since $g$ is continuous at $f(a)$, there exists $\eta > 0$ such that $|y - f(a)| < \eta$ implies $|g(y) - g(f(a))| < \varepsilon$.
Since $f$ is continuous at $a$, there exists $\delta > 0$ such that $|x - a| < \delta$ implies $|f(x) - f(a)| < \eta$.
Therefore, $|x - a| < \delta$ implies $|g(f(x)) - g(f(a))| < \varepsilon$.
$\square$
3 Differentiation
The derivative measures the instantaneous rate of change of a function. Its formal definition relies on limits (see Definition 1).
Definition 3 (Derivative)
The derivative of $f$ at $a$, denoted $f'(a)$, is defined as
provided this limit exists.
Theorem 2 (Chain Rule)
If $f$ is differentiable at $a$ and $g$ is differentiable at $f(a)$, then $g \circ f$ is differentiable at $a$ and
Proof.
Define $\phi(k) = \frac{g(f(a) + k) - g(f(a))}{k} - g'(f(a))$ for $k \neq 0$, and $\phi(0) = 0$. Then $\phi$ is continuous at $0$ and
Setting $k = f(a + h) - f(a)$ and dividing by $h$ gives the result.
$\square$
Remark 1
The chain rule can be written in Leibniz notation as $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$, which suggests (correctly) that derivatives behave like fractions under composition.
4 Integration
Integration, in its Riemann form, formalizes the concept of area under a curve.
Definition 4 (Riemann Integral)
Let $f: [a, b] \to \mathbb{R}$ be bounded. The Riemann integral of $f$ over $[a, b]$ is
where $P = \{x_0, x_1, \ldots, x_n\}$ is a partition of $[a, b]$, $\Delta x_i = x_i - x_{i-1}$, $x_i^* \in [x_{i-1}, x_i]$, and $\|P\| = \max_i \Delta x_i$.
Theorem 3 (Fundamental Theorem of Calculus)
Let $f: [a, b] \to \mathbb{R}$ be continuous. Define $F(x) = \int_a^x f(t) \, dt$. Then:
- $F$ is differentiable on $(a, b)$ and $F'(x) = f(x)$.
- If $G$ is any antiderivative of $f$, then $\int_a^b f(x) \, dx = G(b) - G(a)$.
Example 2
Using Theorem 3, we can compute:
| Function $f(x)$ | Antiderivative $F(x)$ | Domain |
|---|---|---|
| $x^n$ | $\frac{x^{n+1}}{n+1}$ | $n \neq -1$ |
| $\frac{1}{x}$ | $\ln|x|$ | $x \neq 0$ |
| $e^x$ | $e^x$ | $\mathbb{R}$ |
| $\sin x$ | $-\cos x$ | $\mathbb{R}$ |
| $\cos x$ | $\sin x$ | $\mathbb{R}$ |
5 Conclusion
We have presented the foundational concepts of mathematical analysis: limits (Definition 1), continuity (Definition 2), derivatives (Definition 3), and integrals (Definition 4).
Key results include the composition theorem for continuous functions (Theorem 1), the chain rule (Theorem 2), and the fundamental theorem of calculus (Theorem 3).
This paper has also demonstrated the capabilities of markdown-academic for mathematical writing, including:
- Theorem, definition, and proof environments
- Numbered equations with cross-references
- Citations and bibliography
- Tables with captions
- Inline and display mathematics
References
[1] A. L. Cauchy, Cours d'Analyse de l'École Royale Polytechnique, Paris: Imprimerie Royale, 1821.
[2] K. Weierstrass, "Zur Theorie der eindeutigen analytischen Functionen," Abhandlungen der Königlich Preussischen Akademie der Wissenschaften zu Berlin, pp. 11–60, 1876.
[3] B. Riemann, "Über die Darstellbarkeit einer Function durch eine trigonometrische Reihe," Abhandlungen der Königlichen Gesellschaft der Wissenschaften zu Göttingen, vol. 13, 1867.
[4] W. Rudin, Principles of Mathematical Analysis, 3rd ed. New York: McGraw-Hill, 1976.
View markdown-academic source for this paper
+++
title = "On the Foundations of Mathematical Analysis"
subtitle = "A Tutorial Introduction with Examples"
authors = ["Joseph R. Quinn"]
institution = "Department of Mathematics"
date = "2026-02-03"
keywords = ["mathematical analysis", "calculus", "limits", "continuity"]
[macros]
R = "\\mathbb{R}"
eps = "\\varepsilon"
delta = "\\delta"
+++
[[toc]]
::: abstract
This paper serves as both a tutorial introduction to fundamental
concepts in mathematical analysis and a demonstration of the
**markdown-academic** document format...
:::
# Introduction {#sec:intro}
Mathematical analysis forms the rigorous foundation of calculus...
The development of analysis in the 19th century, particularly
through the work of Cauchy [@cauchy1821], Weierstrass [@weierstrass1876],
and Riemann [@riemann1867]...
# Limits and Continuity {#sec:limits}
## Definition of Limits {#sec:limits:def}
::: definition {#def:limit}
**Definition 1 (Limit of a Function).** Let $f: D \to \R$ be a function...
$$\lim_{x \to a} f(x) = L$$
if for every $\eps > 0$ there exists a $\delta > 0$ such that...
:::
::: example {#ex:limit}
Consider $f(x) = x^2$. We claim that $\lim_{x \to 2} x^2 = 4$...
:::
## Continuity {#sec:continuity}
::: theorem {#thm:composition}
**Theorem 1 (Composition of Continuous Functions).** If $f$ is
continuous at $a$ and $g$ is continuous at $f(a)$, then...
:::
::: proof
Let $\eps > 0$ be given. Since $g$ is continuous at $f(a)$...
$\square$
:::
# Differentiation {#sec:differentiation}
The derivative measures the instantaneous rate of change of a function.
Its formal definition relies on limits (see @def:limit).
::: definition {#def:derivative}
The *derivative* of $f$ at $a$, denoted $f'(a)$, is defined as
$$f'(a) = \lim_{h \to 0} \frac{f(a + h) - f(a)}{h}$$ {#eq:derivative}
:::
::: theorem {#thm:chain-rule}
**Theorem 2 (Chain Rule).** If $f$ is differentiable at $a$ and
$g$ is differentiable at $f(a)$, then...
$$(g \circ f)'(a) = g'(f(a)) \cdot f'(a)$$ {#eq:chain-rule}
:::
# Integration {#sec:integration}
Using @thm:ftc, we can compute...
| Function $f(x)$ | Antiderivative $F(x)$ | Domain |
|-----------------|----------------------|------------|
| $x^n$ | $\frac{x^{n+1}}{n+1}$| $n \neq -1$|
| $\frac{1}{x}$ | $\ln|x|$ | $x \neq 0$ |
Table: Common integrals {#table:integrals}
# Conclusion {#sec:conclusion}
We have presented the foundational concepts: limits (@def:limit),
continuity (@def:continuity), derivatives (@def:derivative)...
## References
[^cauchy1821]: A. L. Cauchy, *Cours d'Analyse*, 1821.
[^weierstrass1876]: K. Weierstrass, "Zur Theorie...", 1876.
[^riemann1867]: B. Riemann, "Über die Darstellbarkeit...", 1867.