Continuity
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Properties of continuous functions
Equations are omitted for technical reasons - download the original pdf
(1) The sum of any two continuous functions is continuous. (2) The product of any two continuous functions is continuous. (3) The quotient of any two continuous functions is continuous at any value x for which denominator is not zero. (4) The function [Equation goes here - download the original to see it.] is continuous for all values. The function [Equation] is continuous except for [Equation]. (5) Any polynomial function is continuous for all values. (6) A quotient of two polynomial functions is continuous for all values except where the denominator takes the value zero. (7) The composition of two continuous functions is continuous. Proof Let [Equation] be continuous at [Equation] and let [Equation] Let [Equation] be continuous for [Equation] Let [Equation] Given [Equation] we can find [Equation] such that [Equation] Given [Equation]we can find [Equation] such that [Equation] Then [Equation] Hence [Equation]
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Contents of
Continuity
1 Continuity
2 Properties of continuous functions
3 The intermediate value theorem
4 Bounds of continuous functions
5 A continuous function in a closed interval attains its bounds
6 Uniform continuity
7 Uniform continuity theorem
8 Inverse functions
9 Existence theorem for an inverse function
10 Maxima and minima
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