Continuity

Question 1

Define continuity at a point.

Answer 1

Question 2

What are the three requirements for continuity at a point?

Answer 2

Question 3

When is a function continuous from the left? From the right?

Answer 3

Question 4

When is a function discontinuous at a point “a”?

Answer 4

A function f(x) is discontinuous at a point x=a if:

1. f(x) is undefined at x=a (removable discontinuity)
2. The limit of f(x) DNE (non-removable discontinuity)
3. The limit of f(x) as “x” approaches “a” ≠ f(a)

Question 5

What is the definition of continuity over an interval?

Answer 5

Question 6

What is the greatest integer function?

Answer 6

The greatest integer function is noted by [[x]] or int(x) and is defined as [[x]] = n, where “n” is the greatest integer ≤ “x”.

Examples: 1. [[3.5]] = 3 2. [[-3.5]] = -4

Question 7

What is the definition of continuity over a closed interval?

Answer 7

A function f(x) is said to be continuous over a closed interval [a,b] if and only if:

1. f(x) is continuous on (a,b)
2. f(x) as “x” approaches “a” from the right = f(a)
3. f(x) as “x” approaches “b” from the left = f(b)

Question 8

What are the properties of continuity for two functions “f” and “g” at x=a, “c” serving as a constant.

Answer 8

Question 9

What types of functions are continuous?

Answer 9

Question 10

Define a polynomial function.

Answer 10

Question 11

What is the form/definition of a rational function?

Answer 11

Question 12

What functions are continuous over their entire domains?

Answer 12

1. Polynomials are continuous over (-∞,∞).
2. Rational functions are continuous on (-∞,∞) with the following exceptions:
   
   • D(x)≠0, so except at x=c, such that D(c)=0
3. Exponential functions, where a>0 and a≠1 are continuous over (-∞,∞).
4. trigonometric functions are continuous where they are defined
5. see composite functions

Question 13

Over what intervals are sine and cosine functions continuous?

Answer 13

sin (x) and cos(x) are continuous over (-∞,∞).

• sin(-θ) = -sinθ
• sin(nπ) = 0 for any integer “n”

Question 14

Over what intervals are tangent and secant continuous?

Answer 14

tan(x) and sec(x) are continuous for when x≠nπ/2, where “n” is odd.

Question 15

Over what intervals are cotangent and cosecant continuous?

Answer 15

cot(x) and csc(x) are continuous when x≠nπ, where “n” is any integer.

Question 16

When are composite functions continuous?

Answer 16

If “g” is continuous at “c” and “f” is continous at g(c), then the composite function f(g(c)) is continuous at g(c).

Question 17

Explain composite functions.

Answer 17

Question 18

Explain the intermediate value theorem.

Answer 18

Question 19

Explain the end-behavior test for polynomials.

Answer 19

Question 20

What is the definition of a horizontal asymptote?

Answer 20

Question 21

Define a horizontal asymptote?

Answer 21

The line y = L is called a horizontal asymptote if and only if the limit as “x” approaches ±∞ = L

Question 22

Note another theorem.

Answer 22

• If r>0 is a rational number, then the limit of 1/(x^r) as “x” approaches ∞ = 0
• If r>0 is a rational number such that x^r is defined for all “x”, then the limit as “x” approaches “-a” =0

Question 23

What do the highest degrees of rational functions indicate?

Answer 23

For a rational function:

1. If the degree of N(x) > D(x), then there is NO horizontal asymptote.
2. If the degree of N(x) = D(x), then y = k is the horizontal asymptote, where “k” is the ratio of the coefficients
3. If the degree of N(x) < D(x), y = 0 is the horizontal asymptote.