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Unformatted text preview: . Proof. Observe that f is continuous on [0, π ], & f (0) = 1 < 10 < π ² + 1 = f ( π ). By IVT , there exists c in [0, π ] s.t. f ( c ) = 10 . Problem Let f : [0, 1] → be a continuous function & 0 ≤ f ( x ) ≤ 1 . Show that there exists c in [0, 1] s.t. f ( c ) = c . Summary • Functions ― domain & range • Operations on functions ― compositions • Limits (left & right) Limits involving infinity 0 if p < q = a / b if p = q ∞ if p > q q q q p p p x b x b x b a x a x a + ⋅ ⋅ ⋅ + + + ⋅ ⋅ ⋅ + + − − ∞ → 1 1 1 1 lim • The Sandwich Theorem • Continuity...
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 Winter '11
 yap
 Math, Continuity, Polynomials

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