Prove that exponential function is the only continuous funct

Prove that exponential function is the only continuous function such that

Solution

Let f(x) = e^x    and f(y) = e^y    we Know that e^x is a continuous function as it is defined for all values of x

Similarly e ^y is a continuous function for all values of y.

e ^x+y =e ^x* e ^y is a product of two continuous function.

Therefore e ^x+y is a continuous function as well.

Hence EXPONENTIAL FUNCTION is the only continuous function such that f(x+y) = f(x)* f(y)