University lecture notes template — theorems, definitions, exercises (numbered)

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\title{Lecture Notes 3 --- Continuity and Limits}
\author{}
\date{}

\begin{document}
\maketitle

\section{Limit of a function}

\begin{definition}[Limit at a point]
Let $f: \mathbb{R} \to \mathbb{R}$ and $a, \ell \in \mathbb{R}$. We say that $f(x) \to \ell$ as $x \to a$, written
\[ \lim_{x \to a} f(x) = \ell, \]
if for every $\varepsilon > 0$ there exists $\delta > 0$ such that
\[ 0 < |x - a| < \delta \implies |f(x) - \ell| < \varepsilon. \]
\end{definition}

\begin{remark}
The condition $0 < |x - a|$ explicitly excludes $x = a$: the value $f(a)$ does not have to exist or to equal $\ell$ for the limit to exist.
\end{remark}

\begin{example}[Limit of a polynomial]
For any polynomial $p(x)$ and any $a \in \mathbb{R}$, $\displaystyle \lim_{x \to a} p(x) = p(a)$.
\end{example}

\section{Continuity}

\begin{definition}[Continuous function]
A function $f$ is \emph{continuous at $a$} if $\displaystyle \lim_{x \to a} f(x) = f(a)$. It is \emph{continuous on} $I \subset \mathbb{R}$ if it is continuous at every point of $I$.
\end{definition}

\begin{theorem}[Sum, product and quotient]
\label{thm:operations}
If $f$ and $g$ are continuous at $a$, then so are $f + g$ and $f \cdot g$. If, in addition, $g(a) \neq 0$, then $f / g$ is continuous at $a$.
\end{theorem}

\begin{proof}[Proof sketch]
Apply the limit laws termwise. The product case follows from the standard $\varepsilon$--$\delta$ argument with the bound $|fg - f(a)g(a)| \leq |f| \cdot |g - g(a)| + |g(a)| \cdot |f - f(a)|$.
\end{proof}

\begin{important}
Continuity is a \emph{local} property: it is checked one point at a time. A function may be continuous on $(0, 1)$ and discontinuous at $x = 1$.
\end{important}

\begin{theorem}[Intermediate Value Theorem]
\label{thm:ivt}
Let $f$ be continuous on $[a, b]$. For every $y$ between $f(a)$ and $f(b)$, there exists $c \in [a, b]$ such that $f(c) = y$.
\end{theorem}

\begin{exercise}
Use Theorem~\ref{thm:ivt} to prove that $x^3 - x - 1 = 0$ has at least one real solution in $[1, 2]$.
\end{exercise}

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