1. ## Continuity Question

Let F(X) =

5x-4, if x is less than or equal to 2

-4x+b, if x is greater than 2

If is a function which is continuous everywhere, then we must have b=_______?

2. Originally Posted by redraider717
Let F(X) =

5x-4, if x is less than or equal to 2

-4x+b, if x is greater than 2

If is a function which is continuous everywhere, then we must have b=_______?
intuitively, what does continuous mean? it means there are no gaps, which means, we want -4x + b to pick up exactly where 5x - 4 left off.

let $\displaystyle f_1(x) = 5x - 4$ and $\displaystyle f_2(x) = -4x + b$

we want, $\displaystyle f_1(2) = f_2(2)$

(yes, technically, $\displaystyle x = 2$ is not in the domain of $\displaystyle f_2(x)$, but we do this because we want to get arbitrary close to $\displaystyle x = 2$)

3. Given $\displaystyle F(x) = \left\{ {\begin{array}{c} {5x - 4,\;x \le 2} \\ { - 4x + b,\;x > 2} \\ \end{array}} \right.$

You want: $\displaystyle \lim _{x \to 2^ - } F(x) = \lim _{x \to 2^ + } F(x)$