im...really having a hard time understanding continuity.
In an intuitive sense, "continuity" means exactly what you think it means: the function (or, more specifically, its graph) is "continuous"; it's all connected. The technical definition says that, assuming that there is a limit value (that is, assuming that there's a value that the function "should" take on), the function does take on that value (that is, the function is what it ought to be).
What values of a and b make the function continuous everywhere?
F(x) = ax-4/x-2 if x =/= 2 (DOES NOT EQUAL 2)
b if x= 2
Lacking grouping, the function is as follows:
. . . . .\(\displaystyle \mbox{1. }\, F(x)\, =\, \begin{cases}ax\, -\, \dfrac{4}{x}\, -\, 2&\mbox{if }\, x\, \neq\, 2\\b&\mbox{if }\, x\, =\, 2\end{cases}\)
Is this what you meant? Or were there supposed to be grouping symbols -- "(ax - 4)/(x - 2)" -- so the function is like this?
. . . . .\(\displaystyle \mbox{1. }\, F(x)\, =\, \begin{cases}\dfrac{ax\, -\, 4}{x\, -\, 2}&\mbox{if }\, x\, \neq\, 2\\b&\mbox{if }\, x\, =\, 2\end{cases}\)
Either way, when x = 2, what value do you get? For the graph to be connected, what value "should" the function be, under the first rule, when x = 2? What value does "a" have to be, in order for the first rule's y-value to match the second rule's y-value?
If you get stuck, please reply showing your thoughts and efforts so far. Thank you!
