a) What does it mean that a function $f:\mathbb R \to \mathbb R$ is continuous on $c \in \mathbb R$?
b) Show that the function $f: \mathbb R \to \mathbb R$ given by $$ f(x) = \begin{cases} 3, & \text{if $x \in \mathbb N$}; \\ 0, & \text{if $x \notin \mathbb N$}; \end{cases} $$ not continuous $\forall c \in \mathbb N$.
a) A function $f : \mathbb R \to \mathbb R$ is continuous in $c \in \mathbb R $ if
$$\forall \epsilon \gt 0, \quad \exists \delta \gt 0; \quad c \in \mathbb R, \lvert x - c \rvert \lt \delta \implies \lvert f(x) - f(c) \rvert \lt \epsilon $$
b) We know that $\mathbb N$ is not continuous in $\mathbb R$ because they are a set of isolated points.
Let's assume that the function is continuous. We have
$$\lvert x - c \rvert \lt \delta \implies \lvert f(x) - f(c) \rvert = \vert 3 - 3 \vert = 0 \lt \epsilon \quad ? $$
Got stuck here.