Q1: Let $(f_1, f_2, f_3, \ldots)$ be a sequence of functions $f_n: [0,1] \rightarrow \mathbb{R}$. Which implication is correct?

A1: If it is pointwisely convergent, then also uniformly convergent.

A2: If it is uniformly convergent, then also pointwisely convergent.

Q2: Let $(f_1, f_2, f_3, \ldots)$ be a sequence of continuous functions $f_n: [0,1] \rightarrow \mathbb{R}$. Which implication is correct?

A1: If it is pointwisely convergent to $f$, then $f$ is also continuous.

A2: If it is uniformly convergent to $f$, then $f$ is also continuous.

A3: If it is uniformly convergent to $f$, then $f$ is also differentiable.

Q3: Let $(f_1, f_2, f_3, \ldots)$ be a sequence of differentiable functions $f_n: [0,1] \rightarrow \mathbb{R}$. If the sequence is uniformly convergent to $f: [0,1] \rightarrow \mathbb{R}$, is $f$ also differentiable?

A1: No, never!

A2: Yes, always!

A3: In general, it is not.

Q4: Let $(f_1, f_2, f_3, \ldots)$ be a sequence of differentiable functions $f_n: [0,1] \rightarrow \mathbb{R}$ that is pointwisely convergent to $f: [0,1] \rightarrow \mathbb{R}$. Which implication is correct?

A1: If $(f_1^\prime, f_2^\prime, f_3^\prime, \ldots)$ is uniformly convergent, then $f$ is also differentiable.

A2: If $(f_1^\prime, f_2^\prime, f_3^\prime, \ldots)$ is pointwisely convergent, then $f$ is also differentiable.

A3: If $(f_1^\prime, f_2^\prime, f_3^\prime, \ldots)$ is pointwisely convergent, then $f$ is not differentiable.