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 | 1st 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 | 1st Aug, 2016, 11:38: AM