Answer to Question #92831 in Calculus for MGM

Question

For which values of k, is the function f, defined as below, continuous at x = 2 ?
F(x) = 3-kx, 1≤ x is < 2.
(X2/4) -3 ,x ≥ 2
Further, at which other points in [1,∞[ is f continuous, and why?

Answer 1

In order for function F to be continuous at x=2, it is needed that:

$\lim\limits_{x\rarr2} F(x) = F(2)$ which is equivalent to:

3-2k = 2²/4 -3 or 3-2k = 4/4 -3 => 3-2k = 1- 3 => 3-2k = -2 => 2k =5 => k =5/2

Thus, for k=5/2, the function F is continuous at x=2.

Excepting x=2, where k must be 5/2 for the function to be continuous, F is continuous at all points from the provided interval. The observation for x=1 is that the function is continuous on the right of x=1 since

$F(1)=\lim\limits_{x\rarr1} F(x)= 3-k$

(the function is not defined when x <1).

For any of those points from the interval, F is continuous because the value at that point equals the value of the two one-sided (left and right) limits at that point.