Classification of Compact One-Manifolds/Lemma 1

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Lemma for Classification of Compact One-Manifolds

Let $f$ be a function on $\closedint a b$ that is smooth and has a positive derivative everywhere except one interior point, $c$.

Then there exists a globally smooth function $g$ that agrees with $f$ near $a$ and $b$ and has a positive derivative everywhere.

Proof

Let $r$ be a smooth nonnegative function that vanishes outside a compact subset of $\openint a b$, which equals $1$ near $c$, which satisfies:

$\ds \int_a^b r = 1$

Define:

$\ds \map g x = \map f a + \int_a^x \paren {k \map r s + \map {f'} s \paren {1 - \map r s} } \rd s$

where the constant:

$\ds k = \map f b = - \map f a - \int_a^b \map {f'} s \paren {1 - \map r s} \d s$

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$\blacksquare$

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