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.

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