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= ?

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

c = - 4/3