CBSE Class 12-science Answered

If function f(x) = x , if x is rational and 1 - x , if x is irrational ,then the number of points at which f(x) is contineuos , is (a) ∞ , (b) 1 , (c) 0 , (d) none of these

Asked by yeeshuraj3 | 01 Aug, 2016, 10:12: AM

Expert Answer

W e space c a n space s h o w space t h a t space t h e space o n l y space p o i n t space a t space w h i c h space t h e space g i v e n space f u n c t i o n space i s space c o n t i n u o u s space i s space a t space x equals 1 half left parenthesis a s s u m i n g space t h e space i n t e r v a l space o f space f space t o space b e space left square bracket 0 comma 1 right square bracket right parenthesis T o space s h o w space colon thin space f left parenthesis x right parenthesis space i s space c o n t i n u o u s space a t space o n l y space o n e space p o i n t space x equals 1 half left parenthesis a s s u m i n g space t h e space i n t e r v a l space o f space f space t o space b e left square bracket 0 comma 1 right square bracket space right parenthesis L e t space a space b e space a n y space p o i n t space i n space t h e space i n t e r v a l space open square brackets 0 comma 1 close square brackets. space F o r space e a c h space p o s i t i v e space i n t e g e r space n space w e space c a n space f i n d space a subscript n space element of space Q space a n d space a space b element of R minus Q space comma space b o t h space i n space left square bracket 0 comma 1 right square bracket s u c h space t h a t colon space space space space space space space space space space space space space open vertical bar space a subscript n minus space a close vertical bar space less than 1 over n space comma space space space open vertical bar space b subscript n minus space a close vertical bar space less than 1 over n T h e r e f o r e space space space space stack l i m with n rightwards arrow infinity below a subscript n equals a equals stack l i m with n rightwards arrow infinity below b subscript n space I f space f space i s space c o n t i n u o u s space a t space a space comma space t h e n space w e space m u s t space h a v e space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space stack l i m with n rightwards arrow infinity below f left parenthesis a subscript n right parenthesis equals f left parenthesis a right parenthesis equals stack l i m with n rightwards arrow infinity below f left parenthesis b subscript n right parenthesis N o w space comma space f left parenthesis a subscript n right parenthesis equals space a subscript n space f o r space a l l space i n t e g e r s space n space a n d space f left parenthesis b subscript n right parenthesis space equals space 1 minus space b subscript n space f o r space a l l space i n t e g e r s space n. T h e r e f o r e space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space stack l i m with n rightwards arrow infinity below f left parenthesis a subscript n right parenthesis equals space stack l i m with n rightwards arrow infinity below space a subscript n equals a a n d space space space space space space space space space space space space space space space space space space space space space stack l i m with space space space space space space space space space space space space space space space space space n rightwards arrow infinity below f left parenthesis b subscript n right parenthesis equals stack l i m with n rightwards arrow infinity below open parentheses 1 minus b subscript n close parentheses equals space 1 minus a s o space f o r space f space t o space b e space c o n t i n u o u s space a t space a comma space w e space m u s t space h a v e colon space space space space space space space space space space space space space space space space space space a equals f left parenthesis a right parenthesis equals 1 minus a space space space space space space space space space space space space space space space space space space space space space space space space space space space space space a equals 1 half T h u s space a equals 1 half space i s space t h e space n l y space p o i n t space w h e r e space t h e space f u n c t i o n space i s space c o n t i n u o u s space i n space t h e space i n t e r v a l space left square bracket 0 comma 1 right square bracket. s o space t n e space f u n c t i o n space i s space c o n t i n u o u s space a t space o n e space p o i n t. T h e space c o r r e c t space o p t i o n space i s space b space a s space i t space i s space e q u a l space t o space o n e.

Answered by Vijaykumar Wani | 01 Aug, 2016, 11:38: AM

CBSE Class 12-science Answered

Concept Videos

Continuous and Discontinuous function

Concepts related to continuous and discontinuous functi ...

Continuity ...

CALL US / WHATSAPP

Please Login or Sign up to continue!

CBSE Class 12-science Answered

Concept Videos

Continuous and Discontinuous function

Concepts related to continuous and discontinuous functi ...

Continuity ...

Choose Filters

Open In Ask A Doubt App ?

Get in Touch

Call / Whats App