00:00 Intro

00:18 Definition Continuity

01:22 Special case (holes in the domain of definition)

02:56 Extended definition of continuity

03:48 Examples

09:28 Credits

Q1: Which of the following statements is equivalent to the statement: $f: \mathbb{R} \rightarrow \mathbb{R}$ is continuous at $x_0$.

A1: For all convergent sequences $(x_n) \subseteq \mathbb{R} $ with limit $x_0$, we have $\lim_{n \rightarrow \infty} f(x_n) = f(x_0)$.

A2: For all convergent sequences $(x_n) \subseteq \mathbb{R} \setminus{ x_0 }$ with limit $x_0$, we have $\lim_{n \rightarrow \infty} f(x_n) = x_0$.

A3: For all convergent sequences $(x_n) \subseteq \mathbb{R} \setminus{ x_0 }$ with limit $x_0$, we have $\lim_{n \rightarrow \infty} x_n = x_0$.

A4: For all convergent sequences $(f(x_n)) \subseteq \mathbb{R}$ with limit $f(x_0)$, we have $\lim_{n \rightarrow \infty} x_n = x_0$.

Q2: Is the function $ \displaystyle f(x) = \begin{cases} x^2 &, ~~ x > 0 \ 0 &, ~~x \leq 0\end{cases} ~$ continuous?

A1: Yes!

A2: No!

Q3: Is the function $ \displaystyle f(x) = \begin{cases} x^2 &, ~~ x \geq 0 \ 2-x &, ~~x < 0\end{cases} ~$ continuous?

A1: Yes!

A2: No!

Q4: Is the function $ \displaystyle f(x) = \begin{cases} 1 &, ~~ x \in \mathbb{Q} \ 1 &, ~~x \notin \mathbb{Q} \end{cases} ~$ continuous?

A1: Yes!

A2: No!