I am trying to fit some data sets, which I cannot provide because of data privacy issues. The data sets share a similar trend, they seem to suggest that the underlying function is first exponentially decreasing with a time constant $a_0$. Then the values seem to follow a linear curve with slope $a_1$ for an interval between $x_{\text{lower}}$ and $x_{\text{upper}}$ until it starts to grow like an exponential function with a time constant $a_2$.
I tried something like $f(x)=c_1\exp(-x/a_0)+c_2\exp(x/a_2)$ but I fail to capture the linear part in the interval.
Additional constraints: The functions should have no bumps, be non-zero everywhere and all data sets start at the same function value for $x=0$. Ideally, it should also be infinitely times differentiable or at least continuously differentiable once.
Suppose you wanted to fit
$$e(x)=c_1\exp(-x/a_1)+c_2\exp(x/a_2)$$
With the linear line
$$l(x)=mx+b$$
This may be done with the simple approach of
$$f(x)=\frac{f_1e(x)+f_2l(x)}{f_1+f_2}$$
placing weight on which piece you want accordingly. If the linear function is causing too much of an effect for values outside the region you want to be linear, mainly for $x<x_0$, try using
$$l(x)=\frac{mx+b+m\sqrt{(x-x_0)^2+1}}2$$
For example, with $e(x)=\exp(-x/3)+2\exp(x/5)$, $l(x)=\frac{x+2+\sqrt{(x+2)^2+1}}2$, and $f(x)=\frac{e(x)+2l(x)}3$