13.2 day 2 Continuity

Definition of continuity for a function of 2 independent variables
If lim f(x,y) = f(a,b)
(x,y) (a,b)
Then f(x,y) is continuous at (a,b)

Example 5 a
Test for continuity.

The function f is not continuous at (0,0)

Example 5b
Test for continuity:
(answer is on the next slide)

Test for continuity
This graph is not continuous at y=x 2 .
Why are there lines joining the 2 sections of the graph?
Example 5 b

Examples of continuous functions in R 3

1)
Continuity of composition functions
2)
(answer on next slide)

Continuity of composition functions
1) 2)
Problem 2 is guaranteed to be continuous because it is a composition of continuous functions while problem 1 is not guaranteed to be
Continuous (and is not continuous when y = 3/2x)

The definitions for 2 independent variables can be expanded to
3 independent variables by considering the points inside a sphere.

Test for continuity of a function of 3 independent variables
Find any points of discontinuity

Test for continuity of a function of 3 independent variables
Find any points of discontinuity f(x,y,z) is continuous at all points except on the paraboloid z = x 2 + y 2

Problem 60

Problem 42

• Teacher: The first famous paradox was invented by Zeno, a Greek philosopher about 450 BC.
• Student: Why did he invent the paradox?
• Teacher: I think that it was because he had 2 fishing boats.
• By
• Mr. Whitehead