1. Graph the equation y = f(x), where f(x) = x when x < 2 and 2x-a otherwise, for several different values of a.
(On the Y= screen, enter when(x<2,x,2x-a)|a=0. You will need an actual numerical value for a before the calculator will graph anything. Change the value of a several times.)
Which value of a "looks right"?
The function with the "right" value of a is continuous; the functions with the "wrong" values of a are discontinuous at x = 2.
Graph the function you get when a = 0.
Make a reasonable
guess at the definition of "jump discontinuity."
2. Find the value of b that makes g continuous, where g(x) = x2 when x < -1, g(x) = -x3 when x > -1, and g(-1) = b. (Calculator syntax: define g(x)= (when(x<-1,x^2,0) + when(x=-1,b,0) + when(x>-1,-x^3,0))|b=4. Graph g with the "right" choice of b and with at least one wrong one. You might need ZoomDec to see the difference.)
If b has any of the possible "wrong" values, such as b = 4, g is said to have a removable discontinuity at x = -1.
Make a reasonable
guess at the definition of "removable discontinuity."
3. Graph the function defined by the formula h(x) = 1/x.
(This formula only makes sense if x is not equal to 0, so we'll suppose h is only defined for nonzero real numbers. Since 0 is not in the domain of h, h is officially a continuous function, even though it's not continuous at 0.)
If we include 0 in the domain of the function, we get something like k(x) = when(x><0,1/x,c). (You can also get "not equal to" as a single symbol.)
Is there a way to choose a value for the number c so that the function k is continuous at x=0?
Make a reasonable
guess at the definition of "infinite discontinuity."
4. Graph the function F defined by the formula F(x) = when(x><0,sin(1/x),0). Zoom in on the origin to investigate the behavior of F near x = 0.
Make a reasonable
guess at the definition of "oscillating discontinuity."
5. Find a function with a discontinuity of "mixed type." That is, coming from the left you'd use one description and coming from the right you'd use a different one.