Josephood7

Why is the following function discontinuous at x=0x=0?

(a) f(0)f(0) does not exist.
(b) limx→0f(x)limx→0f(x) does not exist (or is infinite).
(c) Both (a) and (b).
(d) f(0)f(0) and limx→0f(x)limx→0f(x) exist, but they are not equal.

Which of the following is a function that has a jump discontinuity at x=2x=2 and a removable discontinuity at x=4x=4, but is continuous elsewhere?

(a) f(x)=2(x−2)(x−4)f(x)=2(x−2)(x−4).

(b) f(x)=⎧⎩⎨1x−33if x≤2if 2<x<4 or x>4if x=4f(x)={1if x≤2x−3if 2<x<4 or x>43if x=4.

(c) f(x)=⎧⎩⎨2−x21x2−4xif x≤2if x>2f(x)={2−x2if x≤21x2−4xif x>2.

If f(x)=x3−x2+xf(x)=x3−x2+x, is there a number cc such that f(c)=10f(c)=10?
Answer "y" for yes or "n" for no below.

Find the value of the constant cc that makes the following function continuous on (−∞,∞)(−∞,∞).

c=c= 

For the functions below that have a removable discontinuity at x=ax=a [if the function does not have a removable discontinuity, type in "n" below], state the value of g(a)g(a), where g(x)g(x) agrees with f(x)f(x) for x≠ax≠a and is continuous everywhere.

(a) f(x)=x2−2x−8x+2f(x)=x2−2x−8x+2, a=−2a=−2

(b) f(x)=x−7|x−7|f(x)=x−7|x−7|, a=7a=7

(c) f(x)=x3+64x+4f(x)=x3+64x+4, a=−4a=−4

(d) f(x)=3−x√9−xf(x)=3−x9−x, a=9

Consider the function f(x)=2x3+2x2+13f(x)=2x3+2x2+13. For what values of kk does the Intermediate Value Theorem tell us that there is a cc in the interval [0,1][0,1] such that f(c)=kf(c)=k?