` x = 0 ` पर फलन ` f ( x ) ` की सांतत्यता का परीक्षण कीजिये यदि
` f ( x ) ={{:((|x|)/(x)",",, "यदि " x ne 0), ( 1",",, "यदि " x = 0 ) :} `

Explanation

फलन की वाम पक्ष सीमा
माना` x = 0 - h ` , यदि ` x = 0 rArr h = 0 `
` therefore lim _ ( x to 0) rArr lim _ ( h to 0 ) `
` therefore lim _ ( x to 0 ^( - ) ) f ( x ) = lim _ (h to 0 ) f ( 0 - h ) `
` = lim _ ( h to 0 ) ( | 0 - h |)/( ( 0 - h ) ) = lim _ ( h to 0 ) ( | - h|)/( -h ) = lim_ ( h to 0 ) ( h ) /( - h) `
` = - 1 `
फलन की दक्षिण पक्ष सीमा
माना ` x = 0 + h `, यदि ` x = 0 rArr h = 0 `
` therefore lim _ ( x to 0) rArr lim _ ( h to 0 ) `
` therefore lim _ ( x to 0 ^(+)) f ( x ) = lim _ ( h to 0 ) f (0 + h ) `
` = lim _ ( h to 0 ) (| 0 + h |)/( ( 0 + h)) = lim _ ( h to 0 ) (|h|)/(h) = lim _ ( h to 0 ) ( h) /(h) =1 `
` x = 0 ` पर फलन का मान ` f ( 0 ) = 1 `
` because ` फलन की वाम पक्ष सीमा `ne ` फलन की दक्षिण पक्ष सीमा = फलन का मान
अतः x = 0 पर फलन सतत नहीं है |