Zeige, daß die folgenden R^P-wertigen Funktionen an jeder...

Zeige, daß die folgenden $R^{P}$-wertigen Funktionen an jeder Stelle ihres jeweiligen Definitionsbereichs stetig sind:
a) $f(x, y):=\left(x y, x^{2}-y^{2}\right)$,
b) $f(x, y):=\left(1, \mathrm{e}^{x}, \mathrm{e}^{y}\right)$
c) $f(x, y):=(x y, \ln x)$

Key Concepts

Continuity in Euclidean Spaces

This concept refers to the idea that a function from one Euclidean space to another is continuous if small changes in the input produce small changes in the output. In formal terms, f: R^n ? R^m is continuous at a point if for every ? > 0, there exists a ? > 0 so that whenever the distance between x and the point is less than ?, the distance between f(x) and f(point) is less than ?.

Componentwise Continuity

For functions that output vectors, continuity is typically determined by checking the continuity of each individual component. This means that if each scalar component function is continuous on its domain, then the overall vector-valued function is also continuous.

Operations Preserving Continuity

In calculus and analysis, basic operations such as addition, multiplication, composition, and scalar multiplication preserve continuity. This property means that if the functions involved in these operations are continuous, then the resulting function after applying any of these operations will also be continuous.

Domain Constraints in Function Continuity

Some functions, like the logarithm, are continuous only on specific intervals or domains where they are defined. Understanding the domain on which a function is continuous is crucial since continuity is considered only for the values for which the function is defined. For example, a function involving the logarithm is continuous only where the argument of the logarithm is positive.

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