(related to Proposition: Continuity of Exponential Function of General Base)

Let \(a > 0\) be a fixed positive real number. Note that the function \(x\mapsto x\cdot \ln (a)\) is continuous, since it is the product of continuous functions : the identity \(x\mapsto x\) and the constant function \(x\mapsto \ln (a)\).

By definition, the exponential function of general base \(\exp_a:\mathbb R\to\mathbb R\) is a composition of the continuous function \(x\mapsto x\cdot \ln (a)\) and the continuous exponential function \(y \mapsto \exp(y)\):

\[\exp_a(x)=\exp(x\cdot \ln(a)).\]

It follows from the compositions of continuous functions on a whole domain that the exponential function of general base \(a\) is also continuous on its domain, i.e. it is continuous on whole \(\mathbb R\) for every fixed real number \(a > 0\).

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  1. Forster Otto: "Analysis 1, Differential- und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983