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\begin{document}
\title{Continuity summary}
\maketitle

\begin{enumerate}
\item Real numbers as ordered set with multiplication and addition

\item \re\ as complete ordered field
\begin{enumerate}
\item Definition of algebraic field
\item Definition of ordered field. Completeness axiom
\item Definition of $\le$
\item Definition of upper bound
\item Definition of ``bounded above''
\item Definition of supremum
\item Uniqueness of supremum
\item Archimedean property. Use of completeness to prove existence of
$\sqrt{2}$
\item Another definition of supremum, in therms of $\epsilon$
\item Supremum of unions of sets
\item Supremum of sums of sets
\item Definition of infimum
\item Proof of existence of infimum on bounded-below sets, by completeness
\end{enumerate}

\item Revision on limits of functions
\begin{enumerate}
\item Definition of limits of functions. One-sided limits
\item Definition of continuity at a point
\item Same again
\item Definition of continuity on an interval
\item Well-behavedness of conitnuity under addition etc.
\item Continuity of polynomials
\item Continuity under composition
\item Differentiability implies continuity
\end{enumerate}

\item Results about sequences proved using completeness, and needed to study continuity
\begin{enumerate}
\item Any monotone bounded sequence converges
\item Bolzano-Weierstrass theorem (two proofs)
\item Any sequence of \re\ has a monotonic subsequence
\end{enumerate}

\item Results about continuous functions on a bounded interval
\begin{enumerate}
\item Any CFOABI is bounded and attains its bounds
\item Special case of IVT (two proofs)
\item IVT
\item Corollary: existence of positive roots of positive reals
\item Corollary: any odd real polynomial has $\ge 1$ real root
\item Corollary: continuous image of a CBI is a CBI
\item Existence of a fixed point of $f:[a,b] \to [a,b]$
\end{enumerate}

\item An inverse function theorem

\item Intervals in \re
\begin{enumerate}
\item Definition of intervals in terms of betweenness
\end{enumerate}

\item Integration
\begin{enumerate}
\item Definition of step function
\item Definition of partition
\item Definition of integral of step functions
\item Equality of integrals of subordinate partitions of a step function
\item Definition of refinement
\item Existence of common refinement of partitions; independence of integral from partition used
\item Set of step functions as a vector space; integration as a linear
function from this space to \re
\item Positivity
\item Biggerness of integral of a bigger function. Indicator functions
\end{enumerate}

\item Integrals of continuous functions
\begin{enumerate}
\item Integral as supremum of set of all integrals of step functions
\item Proof
\item Corollary: Integral lies between smallest and biggest rectangles
\item Corollary: Integral has same area as some $f(\xi)$-height rectangle
\item How not to integrate
\item Cobbling step functions together
\item Integral of sum of step functions is sum of integrals
\item Similarly for integrals
\item Definition of upside-down integral
\item Corollary to 9.8: addition of integrals in any order
\end{enumerate}

\item Fundamental theorem of calculus
\begin{enumerate}
\item FTC mk. I (on indefinite integrals)
\item Definition of an anti-derivative (primitive)
\item Constant difference of anti-derivatives
\item FTC mk. II (on anti-derivative)
\end{enumerate}

\item Applications of FTC
\begin{enumerate}
\item Linearity of integral on continuous functions
\item Corollary: biggerness of integrals of bigger functions
\item Integration by substitution
\item Misuse and correct use of substitution
\item Integration by parts
\item Alternative definition of log
\item Equivalence of definitions of log
\end{enumerate}

\item Taylor's theorem revisited
\begin{enumerate}
\item Taylor's theorem in integral form
\item Example: power series expansion of $\log{(1+x)}$
\end{enumerate}

\item Second MVT for integrals

\item Numerical integration; error analysis
\begin{enumerate}
\item Maximum error of trapezium integration
\item Proof
\end{enumerate}

\item Metric spaces
\begin{enumerate}
\item Definition of continuity for any $f:\re ^n\to\re$
\item Definition of a metric space
\item Continuity at a point for any function between metric spaces
\item Likewise over subset of space
\item Definition of open spherical neighbourhood
\item Translation
\item Definition of an open set
\item Openness of open spherical neighbourhoods
\item Warning: dependence of openness on subspace
\item Continuity in terms of open sets
\item Openness of intersections and unions of open sets
\item Topological equivalence of metrics
\item Transitivity of openness
\item Definition of closedness
\item Warning: a set can be neither open nor closed (or both)
\item Definition of a limit point
\item Definition of closure
\end{enumerate}

\item Compact spaces
\begin{enumerate}
\item Definition of cover and subcover
\item Definition of compactness
\item Heine-Borel theorem (compactness of closed intervals)
\item Boundedness and closedness of compact subspaces
\item Converse
\item Compactness of continuous images of compact sets
\item Corollary: Boundedness of a continuous function on a compact space
\end{enumerate}

\item Connected sets
\begin{enumerate}
\item Definition of connectedness and disconnectedness
\item Partition of a metric space; connectedness as unpartitionability
\item Equivalence of definitions
\item Connectedness of open intervals in \re
\item Connectedness of continuous image of connected space
\item The image of a \re-valued continuous function on a connected space
is an interval.
\end{enumerate}

\end{enumerate}
\end{document}