If `f(x)={{:(,(x^(2))/(2),0 le x lt 1),(,2x^(2)-3x+(3)/(2),1 le x le 2):}` then,

Options

A. discontinuous at x = 1
B. discontinuous at x = 2
C. continuous at x = 1
D. None of the above

Answer

C

Explanation

Here, we have to check the conitnuity of the function at x=1.
For this, we have a verify that LHL = RHL = f(1)
`f(1)=(1^(2))/(2)=(1)/(2)`
`LHL=lim_(x to 1^(-))f(x)=lim_(h to 0)f(1-h)`
`=lim_(h to 0) ((1-h)^(2))/(2)=(1)/(2)`
`RHL=lim_(x to 1^(+))f(x)=lim_(h to 0)f(1+h)`
`=lim_(h to 0)[2(1+h)^(2)-3(1+h)+(3)/(2)]=2-3+(3)/(2)=(1)/(2)`
Thus, `LHL=RHL =f(1)=(1)/(2)`
Hence, function is continuous at x = 1.