# 13.2 Day 2 continuity - Mr. Whitehead's Site

```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=x2.
Why are there lines joining the 2
sections of the graph?
Example 5 b
Examples of continuous functions in R3
Continuity of composition functions
1)
2)
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 = x2 + y2
Problem 60
Problem 42
• Teacher: The first famous paradox was
invented by Zeno, a Greek philosopher