Opener :. a) Identify the following discontinuities. b) Find a function that could describe the graph. c) Name an interval on which all of these functions are continuous. M439—Calculus A. Section 2.3: Continuity Continued. Intermediate Value Theorem. f(b). f(a). b. a.
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a) Identify the following discontinuities.
b) Find a function that could describe the graph.
c) Name an interval on which all of these functions are continuous
Section 2.3: Continuity Continued
If y = f(x) is continuous on a closed interval [a,b], then it takes on every value between f(a) and f(b) (and sometimes a couple extras)
The intermediate value theorem implies that a continuous function will have no breaks (no jumps, holes, or limits whose values approach infinity).
If your car is traveling at some speed, it must have gone every speed between 0 mph and your current speed during your drive.
1) Sums: f + g
2) Differences: f – g
3) Products: f • g
4) Scalar Multiples: k • f (for any real number k)
5) Quotients: f/g (as long as g(c) is nonzero)
6) Compositions: f о g
Show that is everywhere continuous.
Where is continuous? Where is it not?
We notice that g(x) has a “removable” discontinuity.
What does this mean? Which other discontinuities are NOT removable?
How do we remove it?
How do we extend the function to a new function, h(x)?