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    \centerline{\Large Foundations of Analysis}
	
    The foundations of the rigorous study of \emph{analysis}
    were laid in the nineteenth century, notably by the
    mathematicians Cauchy and Weierstrass. Central to the
    study of this subject are the formal definitions of
    \emph{limits} and \emph{continuity}.
	
    Let $D$ be a subset of $\mathbf{R}$ and let
    $f \colon D \to \mathbf{R}$ be a real-valued function on
    $D$. The function $f$ is said to be \emph{continuous} on
    $D$ if, for all $\epsilon > 0$ and for all $x \in D$,
    there exists some $\delta > 0$ (which may depend on $x$)
    such that if $y \in D$ satisfies
    \[ |y - x| < \delta \]
    then
    \[ |f(y) - f(x)| < \epsilon. \]
	
    Seien nun $f$ und $g$ stetige Funktionen auf $D$. Daraus
    folgt, dass die Funktionen $f+g$, $f-g$ und
    $f g$ stetig sind. Falls zus\"atzlich $g$ \"uberall
    ungleich Null ist, so ist auch $f/g$ stetig.
	
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