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Do now as a warm-up:

Do now as a warm-up:. Is there some number a, such that this limit exists?. If so, find the value of a and find the limit. If not, explain why not. 1.4 Continuity. f(c) exists. exists. not continuous at c. not continuous at c. not continuous at c. continuous at c.

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Do now as a warm-up:

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  1. Do now as a warm-up: Is there some number a, such that this limit exists? If so, find the value of a and find the limit. If not, explain why not.

  2. 1.4 Continuity f(c) exists exists not continuous at c not continuous at c not continuous at c continuous at c

  3. f defined at c but not continuous at c f undefined at c not continuous at c f defined at c and they're = continuous at c f defined at c not continuous at c

  4. Defn. A function f is continuous at c if all these hold: f(c) is defined Defn. A function f is continuous on an open interval (a,b) if it is continuous at every point in (a,b).

  5. Classifying Discontinuities Essential or Non-removable Removable or point (Holes) 2 sided limit exists Infinite (vertical asymptotes) at least one of the 1 sided limits don't exist Jump 1 sided limits exist

  6. ex. Evaluate: f(1) Is f continuous at x=1?

  7. ex. Evaluate these limits: f(0) Is f continuous at x=0?

  8. Thm. given: c∈R and L∈R f is a function and Thm. f is continuous on (a,b) f is continuous on [a,b]

  9. ex. Find a and b so that f(x) is everywhere continuous.

  10. Discuss the continuity of:

  11. Thm. f(x) and g(x) are continuous at x=c b∈R These are continuous at c (b)(f(x)) f+g f-g fg f/g (as long as g(c) ≠ 0)

  12. Thm. Continuity of a composite g is continuous at c f is continuous at g(c) f(g(x)) is continuous at c f Is f(g(x)) cont's at... 0? -7/3? -2? -5? -3? g

  13. ex. Discuss the continuity of the composite function f(g(x)) if... and and

  14. ex. Find the domain for each function. Where is each function not cont's? What kind of discontinuity is this?

  15. ex. Find the domain. Where is the function not cont's? What kind of discontinuity is this?

  16. Thm. The Intermediate Value Theorem (IVT) f is continuous on [a,b] m∈R f(a)<m<f(b) [a,b] where f(c)=m This is an existence theorem. It says something exists, but not where. Try it! Draw a cont's function between (a,f(a)) and (b,f(b)). Pick any y value between f(a) and f(b). A horizontal line through that y value has to cross your function. f(a) m f(b) c a b http://www.calculusapplets.com/ivt.html

  17. The IVT holds for these: f is continuous on [a,b] m∈R f(a)<m<f(b) The IVT even holds for this: It says there's at least one c value that works. There could be many! But, if f is not continuous on [a,b], the theorem doesn't tell us anything.

  18. Apply the IVT, if possible, on [0,5] so that f(c)= 11 for the function

  19. Apply the IVT, if possible, on [0,3p/2] for y=sin x so that f(c)=0.

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