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lines changed Original file line number Diff line number Diff line change @@ -518,7 +518,7 @@ \subsection{Relative minima and maxima}
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\begin {defn }
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Let $ S \subset \R $ be a set and
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let $ f \colon S \to \R $ be a function. The function $ f$ is said to have
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- a \emph {\myindex {relative maximum} }\index {minimum !relative}
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+ a \emph {\myindex {relative maximum} }\index {maximum !relative}
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at $ c \in S$ if there exists a $ \delta >0 $
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such that for all $ x \in S$ where $ \abs {x-c} < \delta $
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we have $ f(x) \leq f(c)$ .
@@ -1353,7 +1353,7 @@ \subsection{Taylor's theorem}
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of $ f$ at $ c$ ,
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we mean that there exists a $ \delta > 0 $ such that $ f(x) > f(c)$ for
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all $ x \in (c-\delta ,c+\delta )$ where $ x\not =c$ .
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- A \emph {\myindex {strict relative maximum} }\index {minimum !strict relative}
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+ A \emph {\myindex {strict relative maximum} }\index {maximum !strict relative}
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is defined similarly.
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Continuity of the second derivative is not needed, but the proof is more
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difficult and is left as an exercise. The proof also generalizes
Original file line number Diff line number Diff line change @@ -119,9 +119,9 @@ \subsection{Partitions and lower and upper integrals}
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\inf \, \bigl \{ U(P,f) : P \text { a partition of $ [a,b]$ } \bigr \} .
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\end {align* }
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We call $ \underline {\int }$ \glsadd {not:lowerdarboux}
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- the \emph {\myindex {lower Darboux integral} }\index {Darboux integral!lower} and
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+ the \emph {\myindex {lower Darboux integral} } and
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$ \overline {\int }$ \glsadd {not:upperdarboux} the
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- \emph {\myindex {upper Darboux integral} }\index {Darboux integral!upper }.
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+ \emph {\myindex {upper Darboux integral} }\index {Darboux integral}.
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To avoid worrying about the variable of integration,
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we often simply write
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\begin {equation* }
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