Function l(x) is continuous for all real values of x and therefore has no point of discontinuity. Hence lim l(x) as x approaches -4 = 1 = l(-4). Function h is discontinuous at x = 1 and x = -1.ĭ) tan(x) is undefined for all values of x such that x = π/2 + k π, where k is any integer (k = 0, -1, 1, -2, 2.) and is therefore discontinuous for these same values of x.Į) The denominator of function j(x) is equal to 0 for x such that cos(x) - 1 = 0 or x = k (2 π), where k is any integer and therefore this function is undefined and therefore discontinuous for all these same values of x.į) Function k(x) is defined as the ratio of two continuous functions (with denominator x 2 + 5 never equal to 0), is defined for all real values of x and therefore has no point of discontinuity. The denominator is equal to 0 for x = 1 and x = -1 values for which the function is undefined and has no limits. These gaps or breaks can be easily seen in a graph. Function g(x) is not continuous at x = 2.Ĭ) The denominator of function h(x) can be factored as follows: x 2 -1 = (x - 1)(x + 1). What is Continuity in Calculus A function is continuous when there are no gaps or breaks in the graph. It helps that the function inside the square root has a factored form and that. Answer: Step 1: The first step is to find the domain of the function. Find the intervals of continuity for the function. Learn more about regions of continuity as a function and read. That means the intervals of continuity for f ( x) are (, 2) and ( 2, ). We will also see the Intermediate Value Theorem in this section and how it can be used to determine if functions have solutions in a given interval. At which of the x values are all three requirements for continuity satisfied Answers and explanations All three requirements for the existence of a limit are satisfied at the x values 0, 4, 8, and 10: At 0, the limit is 2. Therefore function f(x) is discontinuous at x = 0.ī) For x = 2 the denominator of function g(x) is equal to 0 and function g(x) not defined at x = 2 and it has no limit. A region of continuity is where you have a function that is continuous and is a critical understanding in calculus and mathematics. Continuity In this section we will introduce the concept of continuity and how it relates to limits. $$f(x)$$ is continuous on the closed interval $$$$ if it is continuous on $$(a,b)$$, and one-sided continuous at each of the endpoints.A) For x = 0, the denominator of function f(x) is equal to 0 and f(x) is not defined and does not have a limit at x = 0. With one-sided continuity defined, we can now talk about continuity on a closed interval. One-sided continuity is important when we want to discuss continuity on a closed interval. Otherwise, a function is said to be discontinuous.
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