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\ctrline {\bf FUNDAMENTALS OF ELEMENTARY CALCULUS}
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Many common words are used in mathematics in a specialized way.
Usually the mathematical meaning of a word has some relation to the
common meaning of the word; but mathematical meanings are precise,
whereas common meanings are broad or variable.  The adjective ``continuous''
is a word of this kind, with a restrictive and precise mathematical
meaning.  Experience shows that students tend to read more, in the way
of preconceived notions about the meaning of the term, into the word
``continuous'' than is implied by the definition.  In analytical geometry and
calculus we become familiar with the graphs of many functions, and
there is a tendency to associate the term ``continuous function'' with the
picture of a smooth, unbroken curve.  Now it is true that if $f$ is continuous
at each point of an interval, the corresponding part of the graph of
$y=f(x)$ will be an unbroken curve.  But it need \bf not \rm be smooth.  Smoothness
is related to differentiability; the more derivatives $f$ has the smoother
is its graph.  A function may be continuous without having a derivative.
In that case the graph of $y=f(x)$ might be so {\sl crinkly}, so devoid of smoothness,
as to make the correct visualization of it quite impossible.

If $f$ is defined throughout some interval containing $x↓0$ and all points
near $x↓0$, $f$ is continuous at $x↓0$ if to each positive $\epsilon$ corresponds some
positive $\delta$ such that
$$|f(x)-f(x↓0)|<\epsilon \quad \textstyle{whenever} \quad |x-x↓0|<\delta. \eqno(1.1-9)$$

\noindent This form of the condition for continuity is equivalent to the original
definition.

By a rational function of $x$ we mean a function defined by an expression
$$R(x)={p(x)\over P(x)}$$

\noindent where $p(x)$ and $P(x)$ are polynomials.  The function is defined except
when $P(x) = 0$.

$$\int↓a↑b f(x)dx= \lim \sum↓{i=1}↑m f(x↓i↑\prime )\Delta x↓i$$

$${d \over dx} \left[ f(x) \over g(x) \right] = {g(x)f↑\prime (x)-f(x)g↑\prime (x) \over [g(x)]↑2}.$$

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