Interchange.pdf - REAL ANALYSIS INTERCHANGE THEOREMS...

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REAL ANALYSIS INTERCHANGE THEOREMS Theorem 1 Suppose f 12 ( xy ) is continuous at ( ab ) and δ > 0 such that f 2 ( xb ) exists for | x - a | < δ . Then f 21 ( ab ) exists and = f 12 ( ab ). Proof Without loss of generality we may take ( ab ) = (0 , 0). Let ε be given. Choose δ , such that 0 < δ 1 < δ and | f 12 ( xy ) - f 12 (00) | < ε whenever | x | < δ 1 and | y | < δ 1 . Suppose 0 < | h | < δ 1 . Consider f 2 ( ho ) - f 2 (00) h = lim k 0 Δ hk P hk Δ hk = { f ( hk ) - f ( h 0) } - { f ( ok ) - f (00) } We regard k as being temporarily fixed with | k | sufficiently small, and write F ( h ) = f ( hk ) - f ( h 0) so that Δ hk hk = F ( h ) - F (0) hk = F 0 ( θh ) k by MVT0 < θ < 1 = f 1 ( θh, k ) - f 1 ( θh, 0) k = f 12 ( θhθ 0 k ) my MVT0 < θ 0 < 1 Hence fl fl fl fl Δ hk hk - f 12 (00) fl fl fl fl < ε. Letting k 0 we have by (1) fl fl fl fl fl f 2 ( h 0) - f 2 (00) h - f 12 (00) fl fl fl fl fl ε. Hence f 21 (00) exists and is equal to f 12 (00) In the following results R de3notes the closed rectangle a x b c y d . 1
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Lemma Let f ( xy ) be continuous on R . Then we have φ ( x ) = d c f ( xy ) dy is continuous on [ ab ]. Proof f ( xy ) is uniformly continuous on R . Hence, given ε > 0 , δ > 0 || f ( P ) - f ( Q ) | < ε d - c whenever P R Q R and | PQ | < δ . Now if x 1 , x 2 are each in [ ab ] and | x 1 - x 2 | < δ : | φ ( x 1 ) - φ ( x 2 ) | ≤ Z d c | f ( x 1 y ) - f ( x 2 y ) | dy < d - c ε d - c = ε Theorem 2 Let f ( xy ) be continuous as a function of y for c y d relative to this interval, for each x with a x b . Suppose that f 1 ( xy ) is
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  • Fall '98
  • pfitz

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