Let $\left \{ f_{n} \right \}_{n=1}^{\infty}$ be a sequence of functions from $\mathbb{R}$ to $\mathbb{R}$, defined by
$$f_{n}\left ( x \right )=\frac{1}{n}\:exp\left ( -n^{2} x^{2}\right ).$$
Then which one of the following statements is true?
- Both the sequences $\left \{ f_{n} \right \}$ and $\left \{{f}'_{n} \right \}$ converge uniformly on $\mathbb{R}$
- Neither $\left \{ f_{n} \right \}$ nor $\left \{{f}'_{n} \right \}$ converges uniformly on $\mathbb{R}$
- $\left \{ f_{n} \right \}$ converges pointwise but not uniformly on any interval containing the origin
- $\left \{{f}'_{n} \right \}$ converges pointwise but not uniformly on any interval containing the origin
