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Published: 23.06.2021  Python Programming I. Python Programming II. AP CS Principles.

## Continuity and Limits

To develop calculus for functions of one variable, we needed to make sense of the concept of a limit, which we needed to understand continuous functions and to define the derivative. Limits involving functions of two variables can be considerably more difficult to deal with; fortunately, most of the functions we encounter are fairly easy to understand. Sadly, no. Example Looking at figure Fortunately, we can define the concept of limit without needing to specify how a particular point is approached—indeed, in definition 2.

Students will be using the concept of a limit to investigate piecewise functions. This is an introduction into continuity. They will also be introduced to the concept of the average value of a function. Students can graph this with or without a calculator, but this problem comes from the NON-graphing calculator section of the AP exam. Students should be able to graph these functions without the use of a graphing calculator. ## Introduction to Limits and Continuity

The concept of the limit is one of the most crucial things to understand in order to prepare for calculus. A limit is a number that a function approaches as the independent variable of the function approaches a given value. In the following sections, we will more carefully define a limit, as well as give examples of limits of functions to help clarify the concept. Continuity is another far-reaching concept in calculus. A function can either be continuous or discontinuous.

In this section, we strive to understand the ideas generated by the following important questions:. In Section 1. In this present section, we aim to expand our perspective and develop language and understanding to quantify how the function acts and how its value changes near a particular point. Throughout, we will build on and formalize ideas that we have encountered in several settings. We begin to consider these issues through the following preview activity that asks you to consider the graph of a function with a variety of interesting behaviors. Use the graph to answer each of the following questions. If the function does not have a limit at a given point, write a sentence to explain why.

Slideshare uses cookies to improve functionality and performance, and to provide you with relevant advertising. If you continue browsing the site, you agree to the use of cookies on this website. See our User Agreement and Privacy Policy. See our Privacy Policy and User Agreement for details. Published on Feb 29, The concept of limit is a lot harder for functions of several variables than for just one. ## Continuous function

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In mathematics , a continuous function is a function that does not have any abrupt changes in value , known as discontinuities. More precisely, a function is continuous if arbitrarily small changes in its output can be assured by restricting to sufficiently small changes in its input. If not continuous, a function is said to be discontinuous.

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We conclude the chapter by using limits to define continuous functions. Limits are used to make all the basic definitions of calculus. It is thus important for us to gain​.

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