 Solution

The exponents are real, as required for this family of real continuous functions. We prove the exponential Chebyshev system property by induction on the number of functions. A single nonzero multiple of an exponential function has no zero, so the base case holds.

Suppose the result holds for $n$ distinct real exponents, and consider a nontrivial combination $F(x)={\sum }_{j=0}^n{c}_j{e}^{{\lambda }_{j}x}$. If it had $n+1$ distinct zeros, so wouldsince the prefactor is everywhere positive. Applying Rolle's theorem between consecutive zeros gives at least $n$ distinct zeros ofThe $n$ exponents in this derivative are distinct. By induction, a nontrivial combination of them has at most $n-1$ zeros. Therefore every coefficient $c_j({\lambda }_j-{\lambda }_0)$ must be zero. Distinctness gives $c_j=0$ for $j\ge 1$. Then $g=c_0$ has a zero only if $c_0=0$, contradicting the original nontriviality. Consequently $F$ has at most $n$ zeros, and