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...