# Proof of Continuity – A split function with exponential functions and a parameter – Exercise 6591

Exercise

Given the function

$$f(x) = \begin{cases} {(a+x)}^{\frac{1}{x}}+1, &\quad x>0\\ 1, &\quad x = 0\\ \frac{a^x+a^{-x}-2}{x^2}, &\quad x<0\\ \end{cases}$$

a> 0 is a parameter. For what values of the parameter the function is continuous?

$$a=\frac{1}{e}$$