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(a) Draw the graph of a function which is continuous at each point in its domain. (b) Draw the graph of a function which is continuous at every point in its domain but is not differentiable at $x=0.$

(a) Any graph without holes or jumps will do.(b) $f(x)=|x|$ is one such example.

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 3

Limits and Continuity

Derivatives

Harvey Mudd College

Baylor University

Idaho State University

Boston College

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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Yeah, In this problem, we have to find in this problem we have to find such a function which is continuous everywhere. But no differentiate able at X is equal to zero. So let us take the absolute value function for this problem. AARP of acts is equal to absolute value of X, which is equal to by definition we have eggs. If plus X ive X is greater than zero zero, the value of the function will be zero is equal to zero. Hip X is equal to zero and minus X he X is less than zero. So by definition of absolute value of x, we have a piece. Wise function is this function says that f of access plus X f of X is equal to plus X when X is greater than zero. So this is a polynomial. This is a fully normal. So we know that every polynomial functions are continuous everywhere, so f of X is equal to plus X when X is greater than zero is a polynomial, so this will be continuous at everywhere. Similarly, f of X is equal to minus X when X is less than zero is again a polynomial again a polynomial. So we can say that f of X is equal to minus access. Also, continue us everywhere. Now we will check the absolute radio of the function is continuous or not an acceptable to zero. So yeah, so let us check the properties of let us Jake The properties of the properties product function to be continuous The first property is the function should be defined at the given point, so f of zero is equal to from the function. When the function the value of the function is zero when X is equal to zero, so f of zero is equal to zero. This is the first property of continuity. Secondly, the second is limit should be exist at that point. So limit X approaches to zero legs. We will find for us the right hand limit So f of let me x approaches to zero from right hand side of f of X is equal to yes, approaches to zero from right hand side are access greater than zero. We take the pose too well plus X, so this will be equal to zero and the left hand limit left hand side limit as X approaches to zero from left hand side. We have minus X from the definition since access less than euro. So we take minus X. So when you apply the limit, we have again zero. It means that the left hand side limit is equal to the right hand side limit. Therefore, we can say that limit ex abroad just 20 f of x exist, and the third and final condition for a function to be continuous is that limit of the function at that point should be equal to the value of the function at that point. So we have calculated both the values which is equal to zero. So we can say that the given function is continuous everywhere. Now we will take the differential ability of the function at X is equal to zero different. She ability at X is equal to zero in order for derivative. In order for a derivative to exist at a point at a wide, we should have the rate of Adam Boyd we should have limit it goes to zero. Air Force X plus edge minus F of X, divided by edge, must exist if a function is said to be differentiable at any point if it exist the following limit If it exists the following limit. So now we were trying to find. Now we're trying to find limit. It goes to zero f of we have to check the differential ability at accessible to zero. So we have zero plus edge minus f of zero divided by edge where where Airport X is equal to absolute value of acts. So this will be called to limit edge approaches to zero F of zero. Plus it will be absolute value of zero plus edge when we plug zero plus it in the function we have absolute value of zero plus h minus F of zero is zero. The value of the function is actually equal to 00 divided by H which is equal to with girls. Absolute value of media limit edge approaches to zero absolute value of H divided by H. Now we'll trying to find the left limit and the right limit. So let us find the right and slide limit that is limit at approaches to zero from right hand side, absolute radio of as divided by H as it is greater than zero. So we take the so we take to us edge by the information of absolute value and that is greater than zero. We have plus edge divided by edge, which is equal to what? Secondly, we will find the left hand side limit, so limit edge approaches to zero from left hand side means that at just less than zero of the function, it's not really off as divided by age. So this will be equal to limit approaches to zero from left hand side. Yeah, I just less than zero. So by definition of absolutely, we have minus age divided by age, So I'm simplifying and applying the limit. We have minus one since. Since the left hand side limit left hand side limit is not equal to right and side limit. Therefore, therefore, the function is not differentiable at X is equal to zero, and this is the required result where we have to find such a function, which is continuous everywhere but not differentiable at accessible to zero. So we have the function. Absolute value of X is such a function, which is continuous everywhere but not differentiable. We can also see the growth of the absolute value of the function is in the grove. There is no hole and no jumps. Our breaks in the grab so we can conclude their dysfunction is continuous everywhere but at actually equal to zero. We have short time. It means that. And at this point, the left hand side limit is not equal to right hand side limit. So this function is not different. Chevelle at X is equal to zero.

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