Suppose we know the definition of rational exponents of positive real numbers. If we define the exponential function y = a x ( a &gt; 0 ) by extending rational exponents to real exponents by the following limiting process: a x = lim r i ( ∈ Q ) → x a r i ,then how can we prove the exponential function is continuous in R?

Question

Suppose we know the definition of rational exponents of positive real numbers. If we define the exponential function   y  =      a    x       (  a  &amp;gt;  0  ) by extending rational exponents to real exponents by the following limiting process:      a    x    =      lim                  r        i            (      ∈      Q      )      →      x            a                  r        i                 ,then how can we prove the exponential function is continuous in R?

Rylan White · Accepted Answer

We need to show that       a          x      +      δ        =      a    x    +  ϵ with   ϵ depending on   δ and goes to 0 as   δ  →  0. Since       a          x      +      δ        −      a    x    =      a    x    (      a    δ    −  1  ) it is enough to focus on the second term.For a=1+h, then by Bernoulli&#039;s inequality,      a    δ    =  (  1  +  h      )    δ    ≤  1  +  h  δ      |        a    δ    −  1      |    ≤      |    a  −  1      |        |    δ      |  So as   δ  →  0,       a    δ    →  1, and       a          x      +      δ        →      a    x  . Other cases such as a&amp;lt;1 can be handled similarly.

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