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1.4 - Continuity and One-Sided Limits

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Scott Hopkins

on 26 March 2013

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Transcript of 1.4 - Continuity and One-Sided Limits

Continuity and One-Sided Limits
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
Types of Discontinuities
(cc) image by nuonsolarteam on Flickr
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
Find and Label any points of discontinuity
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.
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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