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Decide whether or not the function is continuous. If it is not continuous, identify the points at which it is discontinuous.$$f(x)=|x|$$

Continuous

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 3

Limits and Continuity

Derivatives

Oregon State University

Harvey Mudd College

University of Michigan - Ann Arbor

University of Nottingham

Lectures

04:40

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.

30:01

In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (the rate of change of the value of the function). If the derivative of a function at a chosen input value equals a constant value, the function is said to be a constant function. In this case the derivative itself is the constant of the function, and is called the constant of integration.

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in this problem, we have to decide whether or not the function is continuous. If it is not continuous, we have to identify the points at which it is discontinuous. So the human function is f of X is equal to absolute value of acts. So by the information of absolute value of X, we know that absolute value of X is X if X is greater than zero and the value of the function is equal to zero if X is equal to zero and minus X, if X is less than zero, so f of X is equal to X f of X is equal to X for X is greater than zero. This is a polynomial function, so we know that every polynomial function is continuous. So for X is greater than zero F of accessible to access. Continuous. Similarly F of X is equal to minus X. It's again a polynomial function, so it also continuous for X is less than zero. Now we will check whether this function is continuous at X is equal to zero or not. So check the continuity and X is equal to zero because for access greater than zero function is continuous and X is an X is less than zero for access less than zero. The function is also continuous. Now we had to check at X is equal to zero, so we will check the three conditions. Our three properties of the continuity of a function. The first is the function should be defined at that point, so f of zero is equal to from the absolute value function. The value of the function is zero when X is equal to zero. The second property is the second condition is limit should be exist at that point, that is, limit X Approaches to zero f of X exist so a lot. The limit will be exist if the left hand side limit is equal to the right hand side limit. So let us find the left hand side limit fast, so limit X approaches to zero from left hand side F of X is equal to is X approaches to zero from left hand side. It means that X is less than zero. So we have minus X from the function that is minus X. When X is less than Europe, we take this part so on applying limit. We have zero similarly limit ex abroad just 20 that is right ends. I'll limit F of X is equal to S X approaches to zero from right hand side. So as X approaches to zero from right hand side, it means that X is greater than zero. So for X is greater than zero from the function we take this part that is plus X so on, implying on applying limit we have its value is also zero. So the left hand side limit is equal to the right hand side limit. So therefore we can say that limit exists at X is equal to zero, which is also equal to zero and final, and the third property or the third condition is limit. The value of the limit should be equal to the value of the function a dead bank. So we can see that the value of limit and the value of the function at X is equal to zero is zero. Hence, all the properties are satisfied by the given function. So we can say that this function is continuous. This function is continuous at everywhere, every where

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