Uniqueness of solution of differential equation $y'=y$ In school, I have recently been learning about simple differential equations. We know that the solution of $y'=y$  is $y=Ae^x$, where $A$ is a constant. But how can we know that it is the only solution? The only thing I can figure out is that $y$ is continuously differentiable. Help me, please.
Let $z$ be a solution to $y'=y$. Consider $z(t)e^{-t}$. We have
$$
\frac{d}{dt}(z(t)e^{-t})=z'(t)e^{-t}-z(t)e^{-t}=0,
$$
therefore 
$$
z(t)e^{-t}=Const\implies z(t)=Ae^{t}.
$$
Suppose there is another solution $y=f(x)$. Then we must have that $y=f(x)-Ae^x$ is a solution. Now working this out gives us:
$$y'=\frac{\partial }{\partial x}(f(x) - A e^x) = f'(x) - A e^x$$
Note that we must have $y=y'$ so $f(x)=f'(x)$ clearly this only holds for $f(x) = A e^x$.