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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)

11_partial_differentiation-104.gif if st(y) ≠ 0.

(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)


Last Update: 2006-11-05