Match each function in List I with the number of points of the open interval (0, 2) at which it is not continuous (List II). | List I | List II | |---|---| | A. [x] | I. 3 points | | B. [x^2] | II. No point | | C. [x + 1/2] | III. 1 point | | D. [x/2] | IV. 2 points |

Match each function in List I with the number…

Match each function in List I with the number of points of the open interval (0, 2) at which it is not…

Question 81

HardOriginal question

Match each function in List I with the number of points of the open interval $(0, 2)$ at which it is not continuous (List II).

List I	List II
A. $[x]$	I. 3 points
B. $[x^{2}]$	II. No point
C. $\left[x + \frac{1}{2}\right]$	III. 1 point
D. $\left[\frac{x}{2}\right]$	IV. 2 points

A-I, B-III, C-IV, D-IICorrect answerYour answerA-III, B-I, C-II, D-IVCorrect answerYour answerA-I, B-III, C-II, D-IVCorrect answerYour answerA-III, B-I, C-IV, D-IICorrect answerYour answer

Show the answer and explanation

CorrectNot quiteAnswer D

Correct answer: D

$[u]$ jumps exactly where $u$ crosses an integer. For $0 < x < 2$ , $x$ crosses 1, and $x^{2}$ runs over $(0, 4)$ , crossing 1, 2, 3 at $x = 1, \sqrt{2}, \sqrt{3}$ .
$x + \frac{1}{2}$ runs over $\left(\frac{1}{2}, \frac{5}{2}\right)$ and crosses 1 and 2; $\frac{x}{2}$ runs over $(0, 1)$ and crosses none.

Worth noting

$[x^{2}]$ also jumps at the irrational points $\sqrt{2}$ and $\sqrt{3}$ , where $x^{2}$ is an integer.

Concept: Discontinuities of the greatest integer function
Typical timeTypical: 150 s
Your timeYours: —

The rule this testsContinuous functions

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Ten questions from Continuity & Differentiability, at your own pace.

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Ten questions from Continuity & Differentiability, at your own pace.

Match each function in List I with the number of points of the open interval (0, 2) at which it is not…

Question 81

Explanation