In the Principles of Mathematical Analysis, Rudin introduced a concept named finite valued function. Many students get confused with this terminology. Does finite valued mean bounded? The answer is no. Before we introduce this concept, we have to get to know what is extended real line. Extended real line is a concept that introduced by this textbook which is related to real line R\mathbb{R}R.
R=(−∞,+∞)\mathbb{R}=(-\infty, +\infty)R=(−∞,+∞)
and extended real line is
[−∞,+∞][-\infty, +\infty][−∞,+∞]
In another word, the extended real line is
R∪{±∞}\mathbb{R} \cup \left\{ \pm \infty \right\}R∪{±∞}
Now, we can dive into the concept of finite valued function. For example, f(x)=1/xf(x)=1/xf(x)=1/x is a finite valued function on R/{0}\mathbb{R}/\left\{ 0 \right\}R/{0} and it is not bounded on R/{0}\mathbb{R} / \left\{ 0 \right\}R/{0}. However, if we extended the domain of this function into extended real line, the property of this function changes. It is a finite valued function on [0,+∞][0, +\infty][0,+∞] by letting f(0)=+∞f(0)=+\inftyf(0)=+∞, because f(0)=+∞f(0)=+\inftyf(0)=+∞ is not a finite value.
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