By A. F. Bermant
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Additional resources for A Course of Mathematical Analysis, Part II
The proofs of these theorems will be found in more complete works on analysis*. Theorems 1-3 can be carried over word for word to functions of any number of independent variables. II. The rules for passage to the limit in a sum, product and quotient of functions also hold in the case of functions of n independent variables. The following theorems are easily proved with the aid of these rules, as in the case of a single independent variable. THEOREM 1. The sum and product of a finite number of functions which are continuous at a given point are also continuous at this point.
Given any positive 8, a positive c5 can be found such that If(P) - f(P o) I < 8, provided P Po < 15, independently of the position of the point Po in domain D. The proofs of these theorems will be found in more complete works on analysis*. Theorems 1-3 can be carried over word for word to functions of any number of independent variables. II. The rules for passage to the limit in a sum, product and quotient of functions also hold in the case of functions of n independent variables. The following theorems are easily proved with the aid of these rules, as in the case of a single independent variable.
The quotient of two functions continuous at a given point is continuous at this point provided the denominator does not vanish. THEOREM 3. A function of a 'function is continuous if it is compposed of a finite number of continuous functions. Proof. Suppose we have the function of a function u =
A Course of Mathematical Analysis, Part II by A. F. Bermant