Manipal MET2015MathematicsContinuity and Differentiability

If [] denotes the greatest integer function, then f(x)=[x]+ [x+ 1 2 ]

Options

Ais continuous at x= 1 2
Bis discontinuous at x= 1 2
C_ x (1 / 2)⁺ f(x)=2
D_ x (1 / 2)⁻ f(x)=1

Correct answer

B. is discontinuous at x= 1 2

Step-by-step solution

We have, f(x)=[x]+ [x+ 1 2 ]= cases 0, & if 0 x 1 2 1, & if x= 1 2 1, & if 1 2 x 1 cases Clearly, f(x) is discontinuous at x= 1 2 . Also, _ x 1 / 2⁻ f(x)=0 and _ x 1 / 2⁺ f(x)=1

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