Eigenfunction of all shift operators

[montyness]

Prove that if a continuous function $e\left( x \right)$ on $\mathbb{R}$ is eigenfunction of all shift operators, i.e. $e\left( x+t \right) = \lambda_t e\left( x \right)$ for all x and t and some constants $\lambda_t$, then it is an exponential function, i.e. $e\left( x \right)= Ce^{ax}$ for some constants C and a.

Thanks in Advance.

 

[Ray Vickson]

Prove that if a continuous function $e\left( x \right)$ on $\mathbb{R}$ is eigenfunction of all shift operators, i.e. $e\left( x+t \right) = \lambda_t e\left( x \right)$ for all x and t and some constants $\lambda_t$, then it is an exponential function, i.e. $e\left( x \right)= Ce^{ax}$ for some constants C and a.

Thanks in Advance.

Forum rules: show your work.