Suppose is infinitely differentiable on .
(a) If and show that = .
This is the (one) definition of . It is clear that , and all other derivatives of vanish. From this, we can construct a Taylor-like series definition of , and see that it fits with the usual Taylor series of :
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You have to be careful in using this statement, which may not preclude the analyticity of .
Note that The CoffeeMachine's response shows that if f(x)= sin(x), then f satisfies the given conditions. To show the converse, which is the actual problem, you must also show uniqueness for solutions to such a differential equation. Once again, because the Original Poster shows no attempt at a solution, we do not know what he/she has available to do that.