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 \(y \mapsto \exp(y)\):
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 \(a > 0\).