# Democritus was a Greek philosopher who actually developed the atomic theory, he was also an excellent geometer. Democritus was a Greek philosopher who.

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Democritus was a Greek philosopher who actually developed the atomic theory, he was also an excellent geometer. Democritus of Abdera460 370 B.C.

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Most of the techniques of calculus require that functions be continuous. A function is continuous if you can draw it in one motion without picking up your pencil.

Length of Fish

Parking Meter Cost

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Example 1

Example 2

Examples:

The composition of two continuous functions is continuous.

Example 3

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It is continuous at x = 0, x = 3 and x = 4.Take x = 0, since

Hence, by definition f is continuous at x = 0.For the function below, discuss the integer values where y = f (x) is continuous and explain.Example 4123412

y = f (x)

It is continuous at x = 0, x = 3 and x = 4.For the function below, discuss the integer values where y = f (x) is continuous and explain.Example 4123412

y = f (x) Take x = 3, since

Hence, by definition f is continuous at x = 3.12

It is continuous at x = 0, x = 3 and x = 4.For the function below, discuss the integer values where y = f (x) is continuous and explain.Example 4123412

y = f (x) Take x = 4, since

Hence, by definition f is continuous at x = 4.13

For the function below, discuss the integer values where y = f (x) is not continuous and explain.Example 5123412

y = f (x) This function has discontinuities at x = 1 and x = 2.Take x = 1, since

Hence, by definition f is discontinuous at x = 1.

For the function below, discuss the integer values where y = f (x) is not continuous and explain.Example 5123412

y = f (x) This function has discontinuities at x = 1 and x = 2.Take x = 2, since

Hence, by definition f is discontinuous at x = 2.Determine the intervals where y = f (x) is continuous.

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Example 6

JumpInfiniteTypes of Discontinuities:UndefinedRemovable

From the graph of f, state the numbers at which f is discontinuousand describe the type of discontinuity.Example 7

Consider the function

f has discontinuities at .

a) What type of discontinuities occur at x = 1 and x = -1.

SolutionBy definition, x = -1 is a vertical asymptote, infinite discontinuity.

Example 8

Note: The other discontinuity at x = -1 can not be removed, since it is a vertical asymptote.b) Write a piece-wise function using f (x) that is continuous at x = 1.Take x = 1, it follows

Hence, by definition f is continuous at x = 1.

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Example 921

Example 10

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Example 11

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Example 12

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Example 1327