सिद्ध कीजिए कि फलन ` f ( x ) = {{:( (sin^2ax)/(1)"," x ne 0 ), ( 1" ," x = 0):} ` बिंदु ` x = 0 ` पर सतत नहीं है |

Explanation

` f (0-0)= lim_ ( hto0) f( 0 - h) = lim _ (hto 0) sin ^2 a ( - h) `
` = lim _ (hto 0) sin ^2 ah = lim_ (h to 0) a ^(2)h^(2) xx ( sin^(2) ah ) /( a ^(2) h^(2)) `
` =lim _ ( hto 0 ) ( ah ) ^2 * lim _ ( hto 0) ( sin^(2)ah)/((ah)^(2)) `
` = lim_( hto 0) ( ah)^(2) * [ lim _ ( hto 0) ( sinah)/(ah)]^(2) = 0*1 = 0 `
` f (0 + 0) = lim _ ( hto 0) f(0 + h) = lim _ ( h to 0 ) (sin^(2) ah)/(1) = lim _ ( hto 0 ) ( ah) ^(2) * ( sin^(2) ah)/((ah)^(2)) `
` = lim _ ( hto 0 ) a ^(2) h ^(2)lim _ ( hto 0) ( sin ^2ah) /( (ah)^(2)) = 0*1 = 0 ` और `f ( 0 ) = 1 `
अतः `f ( 0 + 0 ) = f ( 0 - 0 ) ne f (0) `