# Ex 5.1 ,7

Last updated at Dec. 8, 2016 by Teachoo

Last updated at Dec. 8, 2016 by Teachoo

Transcript

Ex 5.1, 7 Find all points of discontinuity of f, where f is defined by ๐๏ท๐ฅ๏ทฏ=๏ท๏ท๏ท๐ฅ๏ทฏ+3, ๐๐ ๐ฅโคโ3๏ทฎ โ2๐ฅ, ๐๐โ3<๐ฅ<3๏ทฎ 6๐ฅ+2, ๐๐ ๐ฅโฅ3๏ทฏ๏ทฏ We have, ๐๏ท๐ฅ๏ทฏ=๏ท ๏ท ๏ท๐ฅ๏ทฏ+3, ๐๐ ๐ฅโคโ3๏ทฎ โ2๐ฅ, ๐๐โ3<๐ฅ<3๏ทฎ 6๐ฅ+2, ๐๐ ๐ฅโฅ3๏ทฏ๏ทฏ Case 1 At ๐ฅ =โ3 f is continuous at x = โ 3 if L.H.L = R.H.L = ๐๏ทโ3๏ทฏ & ๐๏ท๐ฅ๏ทฏ = ๏ท๐ฅ๏ทฏ+3 ๐๏ทโ3๏ทฏ = ๏ทโ3๏ทฏ + 3 = 3 + 3 = 6 Thus, L.H.L = R.H.L = ๐๏ทโ3๏ทฏ โ f is continuous at x = โ 3 Case 2 At ๐ฅ =3 f is continuous at x = 3 if L.H.L = R.H.L = ๐๏ท3๏ทฏ Since L.H.L โ R.H.L i.e. โ 6 โ 20 โ f is not continuous at x = 3 Case 3 Let ๐ฅ =๐, where โ3 < c < 3 ๐๏ท๐ฅ๏ทฏ=โ2๐ฅ f is continuous at x = c if ๏ทlim๏ทฎxโ๐๏ทฏ ๐๏ท๐ฅ๏ทฏ=๐(๐) โด f is continuous at ๐ฅ =๐ & โ3 < x < 3 Thus, f is continuous at each x โ(โ3, 3) Case 4 Let ๐ฅ =๐ where c < โ3) โด ๐๏ท๐ฅ๏ทฏ= ๏ท๐ฅ๏ทฏ+3 f is continuous at x = c (where c < โ3) if ๏ทlim๏ทฎxโ๐๏ทฏ ๐๏ท๐ฅ๏ทฏ=๐(๐) Hence ๏ทlim๏ทฎxโ๐๏ทฏ ๐๏ท๐ฅ๏ทฏ=๐(๐) โ f is continuous at ๐ฅ =๐ (๐<โ3) โ f is continuous at all real numbers less than โ 3. Case 5 If ๐ฅ =๐ & ๐>3 ๐๏ท๐ฅ๏ทฏ= 6x+2 f is continuous at x = c where c > 3 if ๏ทlim๏ทฎxโ๐๏ทฏ ๐๏ท๐ฅ๏ทฏ=๐(๐) Thus ๏ทlim๏ทฎxโ๐๏ทฏ ๐๏ท๐ฅ๏ทฏ=๐(๐) โ f is continuous at ๐ฅ =๐ (๐>3) โ f is continuous at all real numbers greater than 3. Hence, f is discontinuous at only ๐ฅ =3 โ f is continuous at all real numbers except 3. f is continuous at x โRโ๏ท3๏ทฏ

Ex 5.1 ,1

Ex 5.1 ,2

Ex 5.1 ,3

Ex 5.1 ,4

Ex 5.1 ,5

Ex 5.1 ,6

Ex 5.1 ,7 You are here

Ex 5.1 ,8

Ex 5.1, 9 Important

Ex 5.1, 10

Ex 5.1, 11

Ex 5.1, 12

Ex 5.1, 13 Important

Ex 5.1, 14

Ex 5.1, 15

Ex 5.1, 16 Important

Ex 5.1, 17

Ex 5.1, 18 Important

Ex 5.1, 19

Ex 5.1, 20

Ex 5.1, 21

Ex 5.1, 22

Ex 5.1, 23

Ex 5.1, 24

Ex 5.1, 25

Ex 5.1, 26

Ex 5.1, 27

Ex 5.1, 28 Important

Ex 5.1, 29

Ex 5.1, 30 Important

Ex 5.1, 31

Ex 5.1, 32

Ex 5.1, 33

Ex 5.1, 34 Important

About the Author

Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He provides courses for Mathematics from Class 9 to 12. You can ask questions here.