Question

For what value of the constant c is the function f continuous on ( -\infty , \infty ) where

f( t ) =
{t2−cct+8 if t∈(−∞,2) if t∈[2,∞)



c= ?

Answer 1

The function is:

f(t) = t2−c if t∈(−∞,2)

f(t) = ct+8 if t∈[2,∞)

Continuity means that there are not holes or jumps, so the limt of t^2 - c as t approachs to 2 by the left must be equal to the value of f(t) when t =2

Limt (as t -> 2 by the left) of t^2 - c = 2^2 - c = 4 - c

and f(2) = c(2) + 8 = 2c + 8

=> 2c + 8 = 4 - c => 3c = 4 - 8 = -4

c = -4/3

Answer: c = - 4/3