Uniqueness of a General IVP

Question

Consider the IVP $$u'=f(x),$$ where $x\in[a,b], u(a)=\alpha$ and $f$ is continuously differentiable.

I was wondering if a unique solution to the IVP above.

My thoughts:

I was thinking that since $f$ is continuously differentiable and we suppose that $f'$ is bounded, this implies $f$ is also Lipschitz. Hence , if $f$ is Lipschitz and we restrict $f'$to be bounded then the IVP must have a unique solution. Is this true?

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If we do not impose the condition$f'$ is bounded, how else can we determine whether a unique solution exists? Can we conclude this from the domain of $f$?

Answer 1

$ \int_a^x f(t) dt= \int_a^x u'(t) dt =u(x)-u(a) =u(x)-\alpha$. Hence

$$ u(x)= \alpha +\int_a^x f(t) dt.$$