f(c) = 0 f(x) = x2 -6x + 8 f(0) = 8 f(3) = -1 0 is between -1 and 8.I'd be interested to know if you find helpful the account in my book Topology and Groupoids (T&G). Verify that the Intermediate Value theorem applies on and find the value of 'c' guaranteed by the theorem. Solution: This function is continuous on the interval (and everywhere else for that matter), and f(-2) =12 f(-1) = -4 Therefore somewhere between x = -2, and x = -1, f(x) passes through 0.Ģ5 Try This For the function f(x) = x2 -6x + 8, a) use the Intermediate Value Theorem to show that the function has a zero on the interval b) find the value of 'c' guaranteed by the theorem such that f(c) = 0.Ģ6 Solution f(0) = 8 f(3) = -1 0 is between -1 and 8. Without graphing, show that f(x) = 2x4 -3x2 + 5x +2 has at least one zero between -2 and -1. If a function is continuous between a and b, then it takes on every value between and Because the function is continuous, it must take on every y value between f(a) and f(b)Ģ3 Here is a formal statement of the Theorem: It could not have skipped over that number. Then there is some point in time, when it was being filled that it held exactly 15,000 gal. The intervals on which f is continuous are (-, 2] and [2, )Ģ1 Suppose this pool holds 22,000 gal. Solution: f(x) has a non – removable discontinuity at x = 2.

Solution: The function is continuous on the interval [-3, )ġ9 Try This: Find the interval on which f is continuous. Section 2 Finding the Intervals on Which a Function is Continuousġ8 Examle 2: Describe the interval(s) on which f is continuous. Examples: y = 3x + cos(x)ġ7 Finding the Intervals on Which a Function is Continuous

Also: Composites of continuous functions are continuous. Otherwise, there is a (non-removable) discontinuity at this point.Ĭontinuous functions can be added, subtracted, multiplied, divided and multiplied by a constant, and the new function remains continuous. If the f(x) values are equal, the function is continuous. (Recall that vertical asymptotes occur where numerator is nonzero and the denominator is zero.) Root functions are continuous, except at x-values that would result in a negative value under an even root For piecewise functions, find the f(x) values at the x-value separating the regions of the function. “Removable” discontinuity if factoring and canceling “removes” the zero in the denominator “Non-removable” otherwise. Polynomials are everywhere continuous Sine and Cosine are everywhere continuous Rational functions and other trig functions are continuous except at x-values where their denominators equal zero. (You can fill the hole.) Nonremovable Discontinuities: infinite oscillating jumpĬontinuous or discontinuous? If discontinuous Removable or nonremovable discontinuity? At what x-value is the discontinuity? What condition is not met? (See slide 4 for conditions) Condition 1: f(c) is defined - means the function exists when x = c Condition 2: limit as x approaches c of f(x) existis – means that the “two sides of the functions meet at c,” no “jumps, or asymptotes Condition 3: Limit and value are equal – means the is no “hole” with a “dot” filled in elsewhere Function has a discontinuity at x = 2 because does not exist (Definition of Continuity Condition 2).ġ1 Example 1d Solution The function is continuous on the entire real line.ġ2 Try These Discuss the continuity of each function. Therefore, the limit as x approaches 2 does not exist. f(x) is continuous on Ĩ Example 1a Solution There is a removable discontinuity at x = 1 because f(1) is not defined (Definition of Continuity Condition 1).ĩ Example 1b Solution The function has a nonremovableĭiscontinuity at x = 0 because does not exist (Definition of Continuity Condition 2).ġ0 Example 1c Solution The limit from the right of x = 2 does not equal the limit from the left. f(x) is continuous on (-3,2)Ī function f is continuous on the closed interval if it is continuous on the open interval (a, b) and The function is continuous from the right at a and continuous from the left at b. A function that is continuous on the entire real line is everywhere continuous. 3.Ī function is continuous on an open interval (a, b) if it is continuous on each point in the interval. 3 Most of the techniques of calculus require that functions be continuous.Ī function is continuous if you can draw it in one motion without picking up your pencil.Ī function is continuous at a point c if the following three conditions are met.

Understanding continuity calculus