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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) (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 = x2 + y2 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