1.4 - ContinuityObjectives: You should be able to… 1. Determine if a function is continuous. 2. Recognize types of discontinuities and ascertain how to remove discontinuities if possible. 3. Apply the Intermediate Value Theorem to continuous functions.

TEST for Continuity/Definition

• f (x) is continuous at the point x = c if and only if ALL 3 of the following hold:

1. f(c) is defined. (closed circle)

2. exists.

3. (closed circle at limit)

**If any of these three fail, then the continuity at x = c is “destroyed!”

Examples of Discontinuity:

1.

2.

3.

Continuity at Endpoints of a Graph on a Closed Interval

• f(x) is continuous at its left endpoint a if…

• f(x) is continuous at its right endpoint b if…

Example:

• Greatest Integer Function:

• At what points is this graph discontinuous?

RemovableDiscontinuities

• a discontinuity at x = c that can be eliminated/removed by appropriately defining (or redefining) f (c).

• (Open circle where limit exists)

• Ex.

Non-RemovableDiscontinuities

• a discontinuity that can not be removed at x = c even if you attempt to redefine the value of f (c).

• Classifications of Non-Removable Discontinuities—Jump, Infinite, and Oscillating (and vertical asymptotes).

Example:

• Discuss the continuity of .

• Discuss the continuity of . (Graph.)

Example:

• Find the limit (if it exists). Discuss the continuity of the graph.

a.

b.

Example:

• Find the limit (if it exists). Discuss the continuity of the graph.

c.

Example:

• Find the x-values (if any) where f(x) is not continuous. Label as removable or non-removable discontinuities.

a.

b.

Example:

• Find the constant, a, so that f(x) is continuous on the entire real line.

Intermediate ValueTheorem (IVT)

• Consider f (x) is a continuous function on [a, b]. If y0 is between f (a) and f (b), then y0 = f (c) for some .

• Graphically:

Intermediate ValueTheorem (IVT)

• Real-life example:• A common sense example: a person’s height.

Suppose a girl is 5 ft tall on her 13th birthday and 5 ft 7 in. tall on her 14th birthday. For any height h between 5 ft and 5 ft 7 in., there has to be a time when her height was exactly h. For ex., there has to be a time between her 13th and 14th birthday that she was 5 ft 4 in. This is reasonable because a person’s height is continuous and does not abruptly change from one value to another.

Example: No Calc

• Use the IVT to show that the polynomial function has a zero on the interval [0, 1]. That is, show that for some value of c, f (c) = 0.

• For the interval [x1, x2], f(c) has to be between f(x1) and f(x2). f(x1) < f(c) < f(x2) or f(x2) < f(c) < f(x1)

Example:

• Verify that the IVT applies to the indicated interval and find the value of c guaranteed by the theorem.

, [0, 3], f(c) = 4