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Want to cite, share, or modify this book? This book uses the So we need to explore the three conditions of continuity at the boundary points of the piecewise function. Any discontinuity would be at the boundary points. To determine the real numbers for which a piecewise function composed of polynomial functions is not continuous, recall that polynomial functions themselves are continuous on the set of real numbers. A piecewise function may have discontinuities at the boundary points of the function as well as within the functions that make it up. Here, we will analyze a piecewise function to determine if any real numbers exist where the function is not continuous. Now that we can identify continuous functions, jump discontinuities, and removable discontinuities, we will look at more complex functions to find discontinuities. Determining the Input Values for Which a Function Is Discontinuous There are breaks in the function’s graph for this 24-hour period, points at which the price of parking jumps instantaneously by several dollars.ĭetermine whether the function f ( x ) = 9 − x 2 x 2 − 3 x f ( x ) = 9 − x 2 x 2 − 3 x is continuous at x = 3. There are real numbers between 12 and 16 that the function never outputs. We can never be charged $13, $14, or $15. Park an additional hour and the charge is $16. Park for two hours and five minutes and the charge is $12. Suppose a parking garage charges $4.00 per hour or fraction of an hour, with a $25 per day maximum charge. Let’s create the function D, D, where D ( x ) D ( x ) is the output representing cost in dollars for parking x x number of hours. If temperature represents a continuous function, what kind of function would not be continuous? Consider an example of dollars expressed as a function of hours of parking. Temperature as a function of time is an example of a continuous function. A function that has no holes or breaks in its graph is known as a continuous function. At no point did the temperature cease to exist, nor was there a point at which the temperature jumped instantaneously by several degrees. There are no breaks in the function’s graph for this 24-hour period. This means all real numbers in the output between 96 ∘ F 96 ∘ F and 118 ∘ F 118 ∘ F are generated at some point by the function according to the intermediate value theorem, In fact, any temperature between 96 ∘ F 96 ∘ F and 118 ∘ F 118 ∘ F occurred at some point that day.
and 4 p.m., the temperature outside must have been exactly 110.5 ∘ F. the temperature had risen to 116 ∘ F, 116 ∘ F, and by 4 p.m. The graph in Figure 1 indicates that, at 2 a.m., the temperature was 96 ∘ F 96 ∘ F. Let’s consider a specific example of temperature in terms of date and location, such as June 27, 2013, in Phoenix, AZ. Determining Whether a Function Is Continuous at a Number In this section, we will investigate functions with and without breaks. This single observation tells us a great deal about the function. We could trace the graph without picking up our pencil. When we analyze this graph, we notice a specific characteristic.
Figure 1 Temperature as a function of time forms a continuous function.
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