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Home  Partial Differentiation Contionuous Functions of Two or More Variables Definition |
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Definition
The definition of a continuous function in two variables is similar to the definition in one variable. DEFINITION A real function f(x, y) is said to be continuous at a real point (a, b) if whenever (x, y) is infinitely close to (a, b),f(x, y) is infinitely close to f(a, b). In other words, if st(x) = a and st(y) = b, then st(f(x, y)) = f(a, b). Figure 11.2.2 shows (a, b) and f(a, b) under the microscope.
Figure 11.2.2 Remark: It follows from the definition that if f(x, y) is continuous at (a, b), then f(x, y) is defined at every hyperreal point infinitely close to (a, b). In fact, it can even be proved that f(x, y) is defined at every point in some real rectangle a1 < x < a2, b1< y < b2 containing (a, b).
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Last Update: 2006-11-05