For what value of the constant c is the function f continuous

Question

For what value of the constant c is the function f continuous on (-infinity,+infinity) when

f(x)= {s²-c      if s E (-infinity,8)

         {cs+4     if s E [8,infinity)

 

Answer 1

A function is continuous if its graph is a connected curve with no jumps, gaps, or holes.  Hence for the function f(s) to be continuous at the point s=8 where the two separate expressions (s²-c and cs+4) meet, we must have:

 

s² - c = cs + 4

 

Otherwise there will be a gap or hole at s=8.  Plugging in s=8, we get:

 

(8)² - c = 8c + 4

 

Solve for c.