| The ebook Elementary Calculus is based on material originally written by H.J. Keisler. For more information please read the copyright pages. |
|
Home  Partial Differentiation Contionuous Functions of Two or More Variables Theorem 1 |
|
 Search the VIAS Library | Index
|
  |
Theorem 1
Here is a list of important continuous functions of two variables. THEOREM 1 The following are continuous at all real points (x, y) as indicated. (i) The Sum Function f(x, y) = x + y. (ii) The Difference Function f(x,y) = x - y. (iii) The Product Function f(x,y) = xy. (iv) The Quotient Function f(x,y) = x/y, (y ≠ 0). (v) The Exponential Function f(x, y) = xy, (x > 0). (i)-(iv) follow at once from the corresponding rules for standard parts, st(x + y) = st(x) + st(y) st(x - y) = st(x) - st(y) st(xy) = st(x)st(y)
(v) is equivalent to the new standard parts rule st(xy) = st(x)st(y) if st(x) > 0. We prove this rule using the fact that eu and ln u are continuous functions of one variable. st(xy) = st(ey ln x) = est(y ln x) = est(y)st(ln x) = est(y) ln st(x) = st(x)st(y)
|
|
| Home Partial Differentiation Contionuous Functions of Two or More Variables Theorem 1 |
|
if st(y) ≠ 0.
Last Update: 2006-11-05