Q1: Let $(X,\mathcal{T})$ be a topological space and $f: X \rightarrow X$ be a map given by $f(x) = x$ for all $x \in X$. Is the map continuous?

A1: Yes, always!

A2: No, never!

A3: It depends on the topology $\mathcal{T}$.

Q2: Let $(X,\mathcal{T})$ be the discrete topological space. What is always correct for every map $f: X \rightarrow X$?

A1: $f$ is continuous.

A2: $f$ is not sequentially continuous.

A3: $f$ is constant.

Q3: Let $(X,\mathcal{T}_X)$ and $(Y,\mathcal{T}_Y)$ be two topological spaces and $f: X \rightarrow Y$ be a continuous map. Is $f$ also sequentially continuous?

A1: Yes, always!

A2: No, never!

A3: Only if $(X,\mathcal{T}_X)$ is given by a metric space.

A4: Only if $(Y,\mathcal{T}_Y)$ is a second-countable space.