U of T · Analysis 1A · Fall

MAT158H1: calculus, made rigorous.

The first half of U of T's theoretical analysis sequence and a mandatory core course for the Mathematics Specialist. Same calculus topics as MAT137 — proven fully, from the completeness of $\mathbb{R}$ up.

About this course

The hard analysis track.

MAT158H1 is "Analysis 1A": proofs and techniques covering elementary logic, limits, continuity, least upper bounds, the intermediate and extreme value theorems, derivatives, the mean value theorem, the inverse function theorem, integrals, the fundamental theorem, and the elementary transcendental functions. This guide uses Spivak's Calculus as a rigorous companion. Best prepared for by working through MAT137 over the summer — see the MAT137 guide.

01

Chapters

In course order. Each page has definitions, theorems, worked proofs, pitfalls, and practice.

MAT158H1Analysis 1A · Fall MAT159H1Analysis 1B · Winter (sequel) Natural pairMAT240H1 Algebra I

  1. Logic, sets & proof

    Statements, quantifiers, and proof methods.

    Open →

  2. The real numbers & least upper bounds

    Field and order axioms; completeness and $\sup$.

    Open →

  3. Limits

    The ε–δ definition and limit theorems, rigorously.

    Open →

  4. Continuous functions

    Continuity at a point and on intervals.

    Open →

  5. The three hard theorems

    Intermediate value, boundedness, extreme value.

    Open →

  6. Derivatives

    The derivative as a limit; the rules, proved.

    Open →

  7. The Mean Value Theorem

    Rolle, the MVT, and consequences.

    Open →

  8. Inverse functions

    The inverse function theorem in one variable.

    Open →

  9. The Riemann integral

    Upper and lower sums; integrability.

    Open →

  10. The Fundamental Theorem of Calculus

    Both forms, with proofs.

    Open →

  11. Transcendental functions

    Logarithm, exponential, and trig, defined rigorously.

    Open →
02

Textbook Spine

The public U of T calendar gives the official topic list, but not a year-specific textbook. This guide uses Spivak's Calculus as the companion text because its proof style matches the course: definitions first, theorem proof second, computation last. Always verify the current instructor's assigned text on Quercus or the syllabus.

Unit-to-textbook alignment
Foundations

Units 1-2 pair with Spivak's opening work on logic, functions, number systems, order, and the least-upper-bound property. Read for proof grammar: how examples become definitions and how definitions become theorems.

Limits and continuity

Units 3-5 pair with Spivak's development of limits, continuous functions, and the hard interval theorems. Treat the Intermediate Value Theorem, boundedness theorem, and Extreme Value Theorem as the central proof targets of the first half.

Differentiation and integration

Units 6-11 pair with Spivak's derivative chapters, inverse functions, Riemann integration, the Fundamental Theorem of Calculus, and the rigorous construction of elementary transcendental functions.

READ

Read after lecture

Use the local unit first for orientation, then read the matching Spivak section slowly enough to reconstruct every omitted algebra step.

PROVE

Rebuild theorem proofs

For each theorem, close the book and write the dependency chain: which definitions and previous results are being used?

DRILL

Mix routine and proof work

Do computational exercises only after the proof setup is clear. MAT158 tests whether computation is justified, not just whether it is correct.

CHECK

Use the calendar topics

The local chapter order follows the official topic list: elementary logic, limits, continuity, least upper bounds, interval theorems, derivatives, inverse functions, integrals, FTC, and transcendental functions.

03

Practice Plan

Use the unit pages for guided questions, then graduate to textbook-style proof sets. A good MAT158 week has four layers.

01

Definition translation

Write every claim in quantifier form before solving. This is mandatory for limits, continuity, differentiability, integrability, and suprema.

02

Theorem reconstruction

Redo one major proof each week without notes: least upper bound, squeeze, IVT, EVT, MVT, inverse function theorem, or FTC.

03

Spivak-style exercises

Choose a balanced set: two routine checks, two proof problems, one counterexample, and one problem where you must decide which theorem applies.

04

Error correction

Rewrite failed solutions in full sentences. The target is not only the answer; it is a proof a grader can follow line by line.

Course-wide challenge bank

These are not replacements for textbook exercises. They are checkpoint problems that test the proof muscles MAT158 uses most.

Challenge 1 · Supremum

Let $A=\{x\in\mathbb{R}:x^2<2\}$. Prove that $\sup A=\sqrt2$.

Solution sketch
First $\sqrt2$ is an upper bound: if $x>\sqrt2$ then $x^2>2$. If $u<\sqrt2$, choose $x$ with $u<x<\sqrt2$; then $x^2<2$, so $x\in A$ and $u$ is not an upper bound. Thus $\sqrt2$ is the least upper bound.
Challenge 2 · Limit proof

Prove $\lim_{x\to a}x^2=a^2$ directly from the definition.

Solution sketch
$|x^2-a^2|=|x-a||x+a|$. First force $|x-a|<1$, so $|x+a|\le |x-a|+2|a|<1+2|a|$. Then choose $\delta=\min(1,\varepsilon/(1+2|a|))$.
Challenge 3 · IVT

Prove that $x^3+x-1=0$ has a solution in $(0,1)$.

Solution sketch
Let $f(x)=x^3+x-1$. It is continuous. Since $f(0)=-1$ and $f(1)=1$, the Intermediate Value Theorem gives some $c\in(0,1)$ with $f(c)=0$.
Challenge 4 · FTC check

Let $F(x)=\int_1^x \dfrac{1}{1+t^2}\,dt$. Prove $F$ is increasing and compute $F'(x)$.

Solution sketch
The integrand is positive, so if $x<y$ then $F(y)-F(x)=\int_x^y(1+t^2)^{-1}dt>0$. By the Fundamental Theorem of Calculus, $F'(x)=1/(1+x^2)$.
04

Resources

MAT158H1 — official calendar listing
MAT137 guide — summer preparation
MAT137 video library (same topics)

Companion text used by this guide: Spivak, Calculus. The public calendar confirms the MAT158H1 topic list but does not publish a current instructor textbook. Verify the assigned text, edition, and problem numbers against the active syllabus each year.