BCH1(2) - . Proof. Observe that f is continuous on [0, π...

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Continuity Hole Jump
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Pole
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f is continuous at c
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Continuity
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Is f continuous at ‘ p ’?
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A function f : D is continuous on D if f is continuous at each point in D . The following functions are continuous on : polynomials , sin x , cos x , | x | , . The rational function p ( x ) / q ( x ) is continuous on D = { x in q ( x ) 0}. The function ln x is continuous on (0, ). x e
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The Intermediate Value Thm ( IVT ) Let f be a continuous function on [ a , b ] s.t. f ( a ) f ( b ). Then for each z with f ( a ) z f ( b ), there exists c in [ a , b ] s.t. f ( c ) = z . Note. If f ( a ) 0 f ( b ), then f ( x ) = 0 has a solution in [ a , b ] .
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Problem . Let f ( x ) = x ² + 1 + sin x . Prove that there exists c in [0, π ] s.t. f ( c ) = 10 .
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Problem . Let f ( x ) = x ² + 1 + sin x . Prove that there exists c in [0, π ] s.t. f ( c ) = 10
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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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BCH1(2) - . Proof. Observe that f is continuous on [0, π...

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