13 2 day 2 Continuity Definition of continuity

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 5 b 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

Continuity of composition functions 1) 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 = 3/2 x)

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