#### Transcript of 1.4 - Continuity and One-Sided Limits

Continuity and One-Sided Limits

1.4

Definition

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.

A function is continuous at a point if the limit is the same as the value of the function.

Continuity Test

Jump

Removable

Point

Essential

Types of Discontinuities

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Types of Discontinuity

Occurs when the curve breaks at a particular point and starts somewhere else.

Right hand limit does not equal left hand limit

Occurs when the curve has a “hole” because the function has a value that is off the curve at that point.

Limit of f as x approaches x does not equal f(x)

Occurs when curve has a vertical asymptote

Limit dne due to asymptote

Occurs when you have a rational expression with common factors in the numerator and denominator. Because these factors can be cancelled, the discontinuity is removable.

Where do I look for discontinuities?

Rational Expression

Values that make denominator = 0

Piecewise Functions

Changes in interval

Absolute Value Functions

Use piecewise definition and test changes in interval

Step Functions

Test jumps from 1 step to next.

Functions Continuous on their domain

Polynomial

Rational

Radical

Trigonometric

**Examples**

Find and Label any points of discontinuity

**Continuity**

2.4 (Day 2)

Find the values for the variables that make the function continuous

**Discussing Continuity**

Are the following continuous on their domain?

**Intermediate Value Theorem**

**Real World Examples**

If between 7am and 2pm the temperature went from 55 to 70.

At some time it reached 62.

Time is continuous

If between his 14th and 15th birthday, a boy went from 150 to 165 lbs.

At some point he weighed 155lbs.

It may have occurred more than once.

**Examples**

Show that a “c” exists such that f(c)=2 for f(c)=x^2 +2x-3 in the interval [0, 2]

Determine if f(x) has any real roots:

Is any real number exactly one less than its cube?

**Max-Min Theorem for Continuous Functions**

If f is continuous at every point of the closed interval [a, b], then f takes on a minimum value m and a maximum value M on [a, b].

Determine continuity at a point and continuity on an open interval.

Determine one-sided limits and continuity on a closed interval.

Use properties of continuity.

Understand and use the Intermediate Value Theorem.

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