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About the region below the graph of a function. (English) Zbl 1331.54020

The author deals with the problem of when a hypograph \(H_f\) of a function \(f: \mathbb R \to \mathbb R\), defined as \(H_f = \{(x, y)\mid y \leq f (x)\}\), i.e., the set of points lying on or below the graph of \(f\), is connected. In the beginning, two simple examples of functions whose hypographs are disconnected are shown. Also two notions, viz. additive functions and everywhere surjections are recalled, the definitions of which are as follows:
a)
A function \(f : \mathbb R \to \mathbb R\) is said to be additive if it satisfies the Cauchy functional equation \(f (x + y) = f (x) + f (y)\) for all \(x\) and \(y\). In fact, a function \(f\) is additive if and only if it is a linear map when the additive group \(\mathbb R\) is regarded as a vector space over the field \(\mathbb Q\).
b)
A function \(f : \mathbb R \to \mathbb R\) is said to be an everywhere surjection (or a strongly Darboux function) if, for every interval \((a, b)\) and for every \(y\), there is an \(x \in (a, b)\) such that \(f(x)=y\).

In the paper, it is shown that, if \(H_f\) is dense in \(\mathbb R^2\), then \(H_f\) is connected; in particular, the hypograph of any discontinuous additive function is connected, the same being true for an everywhere surjection as well.
Characterizations both of connected hypographs and of pathwise connected hypographs are established as follows:
Theorem : Let \(H_f\) be the hypograph of a function \(f : \mathbb R \to \mathbb R\). The following are equivalent:
1. The set \(H_f\) is connected;
2. There is no \(x_0 \in \mathbb R\) such that \(\lim_{x \to x_0 -} \,\, f (x) = - \infty\) or \(\lim_{x \to x_0 +} f (x) = -\infty\)
3. For every interval \((a, b)\) there is a number \(M\) such that the closure of the set \(\{x \in (a, b)\mid f (x) \geq M\}\) contains \(a\) and \(b\).
Theorem : The following are equivalent:
1. The set \(H_f\) is pathwise connected;
2. There is no \(x_0 \in \mathbb R\) such that \(\lim\inf_{x \to x_0}\,\, f (x) = -\infty\);
3. In every closed and bounded interval \(I\) the function is bounded from below (i.e., there exists a number \(M\) such that \(f (x) \geq M\) for all \(x\) in \(I\)).
Moreover, in this context, simple examples of connected but not pathwise connected plane sets (in fact, with uncountably many path-components) are obtained.

MSC:

54D05 Connected and locally connected spaces (general aspects)
26A30 Singular functions, Cantor functions, functions with other special properties
54G99 Peculiar topological spaces
97I20 Mappings and functions (educational aspects)
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References:

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