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Discuss the discontinuities of$$D(x)=\left\{\begin{array}{ll}0 & \text { if } x \text { is irrational } \\1 & \text { if } x \text { is rational }\end{array}\right.$$

Everywhere discontinuous

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 3

Limits and Continuity

Derivatives

Harvey Mudd College

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.

04:24

Discuss $\lim _{x \rightar…

01:32

01:01

Prove that if $x$ is ratio…

01:28

in this problem of limited community, we have to discuss lim X s approaching towards zero and the funds and a ffx and limit. This is limit X- 80 approaching towards one and the function F of X if if we have given it's a function F of X is equals to x square If access if X is rational, professional and This FX is equal to zero if X is irrational, X. S irrational. So first we are considering lim F of X Where X is approaching towards zero. Any open interval containing zero will contain both rational number and irrational number as the rational number is approaching towards zero. So that means actually approaching towards zero. The value of the fund's a week. We approaching towards zero because the sage X square. So this will be zero square is again zero. So this suggested that limit X- 80 approaching towards zero, is equal to zero. So this argument can be made precise. Usually using delta is equal to under root of X using this term. So this will be here actually under root of epsilon. So within absolute and delta definition. Now we are considering second, which is here. So limit X is approaching towards one and F of X. Any open interval containing one. So here containing one Will contain both rational and irrational numbers as the rational number of approaches towards one. So this will be the value of the phantom say X age approaching towards one. So X square would be again approaching towards one. So as the original number of pledges towards one, however, the function value will approach 20. So this would be uh ambiguity. That means between rational and irrational number. So therefore we can see limit. So this would be F f X. This is does not exist, so is not equal to one or we can say does not exist, does not exist. So this would be saying much clear. So this doesn't exist. The argument can be made precisely as falling. So suppose the limit doesn't exist and is equals to L. So here we have doesn't exist now we are supposing that it exists. So supposed limit exists. So limit exists. So here this will be Lim x age approaching towards one and FFX would be equals 12. Now we are taking steps alone is equals to one divided with four. So this would be one divided with four. Then there would be a delta such that Models of X -1, It's the value between 0 to Delta. The value of the function would be within one Divide. With 4th of L. Considering the rational and irrational values of X separately would require lb simultaneously within one day while with four of one and zero, which is impossible because at the time a function would give you a single value but it is saying that it will be having to value at at at at a single time. So that's why this means that limit does not exist. So Here is the solution for first part and here is the solution for 2nd part.

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