Proof. Let X be a path-connected topological space and let \(f:X\rightarrow Z\) be
continuous integer-valued function on X. If \(x_0,x_1\) are any two points of X, there exists a path \(\gamma :[0,1]\rightarrow X\) such that \(\gamma (0)=x_0\) and \(\gamma (1)=x_1\). But then \(fo\gamma :[0,1]\rightarrow Z\) is continuous integer valued function on [0,1]. But [0,1] is connected, and so \(fo\gamma \) is constant. It follows that \(f(x_0)=f(x_1)\). Thus every integer valued function on X is constant. Therefore X is connected.

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Proof. Let \(x_0\) and \(x_1\) be any two fixed points in \(\mathbb {R}\) and define \(\gamma :[0,1]\rightarrow \mathbb {R}\) by \(\gamma (t)=(1-t)x_0+tx_1\) and since \(\gamma \) is a linear function of t it is continuous. Thus \(\gamma \) is a path in \(\mathbb {R}\) connecting \(x_0\) and \(x_1\) and so \(\mathbb {R}\) is connected.
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