00:00 Intro

00:14 Recalling continuity

01:25 Combining 2 continuous functions

03:07 Composition of functions

05:55 Proof for composition of functions

07:27 Credits

Q1: Let $f: \mathbb{R} \rightarrow \mathbb{R}$ and $g: \mathbb{R} \rightarrow \mathbb{R}$ be continuous at $x_0$. Which statement is not correct in general?

A1: The function $f+g: \mathbb{R} \rightarrow \mathbb{R}$ is also continuous at $x_0$.

A2: The function $f-g: \mathbb{R} \rightarrow \mathbb{R}$ is also continuous at $x_0$.

A3: The function $f \cdot g: \mathbb{R} \rightarrow \mathbb{R}$ is also continuous at $x_0$.

A4: The function $f \circ g: \mathbb{R} \rightarrow \mathbb{R}$ is also continuous at $x_0$.

Q2: Let $f: \mathbb{R} \rightarrow \mathbb{R}$ and $g: \mathbb{R} \rightarrow \mathbb{R}$ be continuous. At which points is the function $f/g$ continuous in general?

A1: At all points $x_0 \in \mathbb{R}$.

A2: At no point $x_0 \in \mathbb{R}$.

A3: At the points $x_0 \in \mathbb{R}$ where $g(x_0) \neq 0$.

A4: At $x_0 = 0$.

Q3: Let $f: \mathbb{R} \rightarrow \mathbb{R}$ and $g: \mathbb{R} \rightarrow \mathbb{R}$ be continuous. Which statement is correct?

A1: $f \circ g$ is not well-defined in general.

A2: $f \circ g$ is a continuous function.

A3: There are some points where $f \circ g$ is not continuous.