If f ( x )= array ll m x+1, & x 2 x+ n , & x> 2 array . , is continuous at x= 2 ,( ~m , n Z ) then

Step-by-step solution

The function f(x)= cases m x+1 & x 2 x+n & x> 2 cases must satisfy the continuity condition at x = 2 . The left-hand limit yields _ x ( 2 )^- f(x) = m ( 2 ) + 1 while the right-hand limit gives _ x ( 2 )^+ f(x) = ( 2 ) + n = 1 + n Equating these limits with the function value f ( 2 ) = m ( 2 ) + 1 produces m ( 2 ) + 1 = 1 + n which simplifies to the continuity condition n = m 2 This relationship corresponds exactly with option D.