Find b such that the following function is continuous in [0,1] : f(x)=0 if x<0.5 and f(x) = x-b for x=0.5

Question

Find b such that the following function is continuous in [0,1] : f(x)=0 if x<0.5 and f(x) = x-b for x≥0.5

Answer 1

The trick to this type of problem is to set the two expressions equal at the x-value where we need continuity:

 

In this case,

  the first expression is f_left(x) = 0

  the second expression is f_right(x) = x-b

  The x-value where we need continuity is x=0.5

 

Solving:

  f_left(0.5) = f_right(0.5)

  0 = 0.5 - b

  b = 0.5