Continuity is "no sudden jumps," made precise with ε–δ. It is the hypothesis that powers the three hard theorems of the next unit — and this unit builds the small lemmas those proofs will quietly rely on.
Almost every useful theorem about functions assumes continuity. Pinning down exactly what it means is what lets those theorems be proved rather than believed.
Continuity is the "small change in input gives small change in output" assumption behind stable control systems, well-posed physics models, and convergent numerical methods.
Bolzano and Cauchy gave the modern definition in the early 1800s. Pathologies soon followed — functions continuous at exactly one point, or continuous exactly at the irrationals — showing how much the precise definition was needed.
$f$ is continuous at $a$ if $a$ is in the domain and $\displaystyle\lim_{x\to a}f(x)=f(a)$. Spelled out: $\forall\varepsilon>0\ \exists\delta>0$ such that $|x-a|<\delta\Rightarrow|f(x)-f(a)|<\varepsilon$. On an interval, $f$ is continuous if it is continuous at every point (one-sidedly at included endpoints).
Note the small change from the limit definition: the clause $0<|x-a|$ is gone, because now $f(a)$ itself must fit the tolerance — trivially true at $x=a$.
$\sqrt{x}$ is continuous at every $a>0$.
If $f,g$ are continuous at $a$, so are $f+g$, $fg$, and (when $g(a)\ne0$) $f/g$. Consequently polynomials are continuous everywhere and rational functions are continuous wherever defined.
These all follow from the limit laws of Unit 3 — no new ε work needed. The one genuinely new proof is composition:
If $g$ is continuous at $a$ and $f$ is continuous at $g(a)$, then $f\circ g$ is continuous at $a$.
If $f$ is continuous at $a$ and $f(a)>0$, then there is $\delta>0$ with $f(x)>\tfrac{f(a)}2>0$ for all $x$ with $|x-a|<\delta$. (Similarly for $f(a)<0$.)
$f(x)=x\sin(1/x)$ for $x\ne0$, $f(0)=0$. Since $|f(x)|\le|x|\to0$, $\lim_{x\to0}f=0=f(0)$: continuous at $0$ (and everywhere else by the algebra and composition theorems).
Let $g(x)=x$ for rational $x$ and $g(x)=0$ for irrational $x$.
Discontinuities come in kinds: removable (limit exists but differs from $f(a)$ or $f(a)$ undefined), jump (one-sided limits exist but disagree), and essential (a one-sided limit fails to exist, as with $\sin(1/x)$ at $0$).
• Continuity at $a$ needs three things: $f(a)$ defined, the limit exists,
and they are equal.
• A removable discontinuity is still a discontinuity until you actually
redefine $f(a)$.
• For $f\circ g$, the outer function must be continuous at $g(a)$ — not at
$a$.
• "Continuous on $[a,b]$" means one-sided continuity at the endpoints; this
matters for the theorems of Unit 5.
Original problems written for this guide in the style of Spivak's exercises. Write a full solution before opening the answer.
Where is $f(x)=\dfrac{x^2-1}{x-1}$ (with $f(1)=2$) continuous?
Prove the sum of two functions continuous at $a$ is continuous at $a$.
Is the Dirichlet function ($1$ on rationals, $0$ on irrationals) continuous anywhere?
Prove: if $f$ is continuous at $a$ and $f(a)\ne0$, then $f$ is nonzero on some open interval around $a$.
Show $g(x)=x$ for $x\in\mathbb{Q}$, $g(x)=0$ otherwise, is continuous at $0$ and nowhere else.
Suppose $f$ is continuous on $\mathbb{R}$ and $f(q)=0$ for every rational $q$. Prove $f(x)=0$ for all $x$.
Prove: if $f$ is continuous at $a$, so is $|f|$.
For which $c$ is $f$ continuous at $1$, where $f(x)=x^2$ for $x\le1$ and $f(x)=cx$ for $x>1$?
Course coverage. This unit is part of the complete MAT158H1 topic map. It should cover the following items before you move on:
Use the current UofT MAT158H1 calendar topics as the coverage spine, and pair lecture notes with a Spivak-style proof-calculus text for extra reading. The exact assigned textbook should still be checked against the instructor syllabus.
The continuity spine is: local limit equality at every point, algebra and composition preserve it, and interval/compactness theorems give global conclusions.
Use these after the main practice set. They are designed to force definition-level reasoning, theorem selection, and mixed application rather than routine calculation only.
Prove $f(x)=3x-2$ is continuous at every $a$.
If $f$ is continuous at $a$ and $g$ is continuous at $f(a)$, prove $g\circ f$ is continuous at $a$.
For what $c$ is $f(x)=x^2$ for $x<1$ and $f(x)=cx+1$ for $x\ge1$ continuous at $1$?
Show the Dirichlet function $1_{\mathbb{Q}}$ is discontinuous at every real number.