I was wondering is it possible to show that there exists a sequence $(f_n)$ of differentiable functions such that ($f_n^{'})$ converges uniformly, but $(f_n)$ does not converge in $\mathbb R$.
I understand that if $(f_n)$ is a sequence of differentiable functions defined on [a,b] in $\mathbb R$, and $(f_n^{'})$ converges on [a,b], $(f_n)$ converges if there exists $x_0$ in [a,b] where $f_n(x_0)$ converges on [a,b].
However, the problem is now my domain is in $\mathbb R$, and not [a,b]. I don't know if I can generalize [a,b] to $\mathbb R$, because convergence usually relies on the domain to exploit the construction of $(f_n)$.