25 - Continuity Continuity of a function means the graph...

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Unformatted text preview: Continuity Continuity of a function means the graph has no breaks, jumps, holes, or gaps. Continuity at a point means the function exists there, that the limit exists there, and that these values are the same. Specifically, Definition: A function is continuous at a number lim If a function is not continuous at a point , we say it is discontinuous at . 4. From the graph of , state the intervals on which is continuous. if 10. use defn to show continuous lim lim lim , as required MTH 173, section 2.5, page 1 16. Why discontinuous at what is what is lim lim : 18. why discontinuous at what is what is lim These are not equal 20. why discontinuous at lim not continuous lim lim lim lim lim DNE MTH 173, section 2.5, page 2 In the last example, note that the definition of continuity, thus Definition: A function lim We can modify the definition of with lim in continuity to "continuous from the right" by replacing lim is continuous from the right at a number lim if and Definition: A function is continuous from the left at a number lim if Consider the greatest integer function : At what real numbers is this function discontinuous? at all integers What type of continuity happens at these points? lim lim thuscontinuous from the right Defn: A function is continuous on an interval if it is continuous at every point in the interval. If the interval includes an endpoint, then at that point the appropriate "continuous from a side" applies there. The function is continuous in this interval. MTH 173, section 2.5, page 3 Thm: If and re continuous at are also continuous at : 1. 4. 2. 5. if and is a number, then the following functions 3. and, as a consequence, every polynomial is continuous everywhere, as is every rational function wherever defined. Many of our familiar function families are continuous within their domains, including: polynomials, rationals, trig, inverse trig, exponentials, logs, and roots. tan 24. sin domain is all reals except The numerator function, sin , and the denominator function, , are each continuous everywhere, and thus their quotient is also continuous everywhere except where the denominator is , i.e., at continuity fails. 34. use continuity to evaluate lim arctan lim arctan arctan MTH 173, section 2.5, page 4 38. where discontinuous? lim but lim so lim lim lim and DNE Discontinuous at both Suppose we want to solve the equation ! If there are real number solutions, we can approximate them from a graph of ! by determining the -intercepts. your calculator does this with the "zero" command on the TI83 or 84, this requires that the "bounds" have a sign change in order to determine the zero. try on the 83/84. the 89 also has a zero command, but it works differently, as it will determine the zero in At any rate, the process the 83/84 uses is an example of the MTH 173, section 2.5, page 5 The Intermediate Value Theorem: Suppose that is continous on the closed interval " and let be any number between and " , where " Then there exists a number in " such that 48. use IVT to show there is root in for Set to : or make function: evaluate at and since , such that , we know there is a number #$ , MTH 173, section 2.5, page 6 ...
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