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Calc_Chap03

# Calc_Chap03 - MATH1131 Mathematics 1A Calculus Chapter 03...

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MATH1131 – Mathematics 1A Calculus Chapter 03 Properties of Continuous Functions Dr. Thanh Tran School of Mathematics and Statistics The University of New South Wales Sydney, Australia 1 1 Combining Continuous Functions 2 The Intermediate Value Theorem 3 The Maximum-Minimum Theorem 2 Continuity at a point (revision) Definition 1 Suppose that f is defined on some open interval containing the point a . We say that f is continuous at a if lim x a f ( x ) exists and lim x a f ( x ) = f ( a ) . Combining Continuous Functions 3 Theorem 2 If the functions f and g are continuous at a, then f ± g is continuous at a fg is continuous at a f / g is continuous at a, provided g ( a ) negationslash = 0 Proof: We look only at continuity of fg . As f and g are continuous at a we have lim x a f ( x ) = f ( a ) and lim x a g ( x ) = g ( a ) . Next lim x a ( fg )( x ) = lim x a ( f ( x ) g ( x ) ) (defn of fg ) = parenleftBig lim x a f ( x ) parenrightBigparenleftBig lim x a g ( x ) parenrightBig (props of limits) = f ( a ) × g ( a ) ( f , g are cts at a ) = ( fg ) ( a ) (defn of fg ) Since the limit of fg as x a is the value of fg at x = a , fg is continuous at a . Combining Continuous Functions 4

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Theorem 3 Let g be continuous at a and let f be continuous at g ( a ) , then f g is continuous at a. Proof: Use rule for composition of limits — for you to do! Combining Continuous Functions 5 Example 4 Let f : R R be given by f ( x ) = braceleftBigg cos ( ax ) for x π bx for x > π . For what values of a and b will f be continuous? From the definition of f we see that f is continuous when x negationslash = π . In fact, since f ( x ) = cos ( ax ) when x < π , and the cosine function is continuous, f is continuous.
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Calc_Chap03 - MATH1131 Mathematics 1A Calculus Chapter 03...

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