5.2 Combinations of Continuous Functions

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5.2.1 Theorem: Algebraic operations on continuous function in c

Let

• $A\subseteq \mathbb{R}$
• $f,g:A\to \mathbb{R}$
• $b\in \mathbb{R}$
• $c\in A$
• $f,g$ continuous at $c$

Then

• $f+g,f-g,fg,bf$ continuous at $c$
• $h:A\to \mathbb{R}$ continuous at $c$, $h(x)\ne 0\forall x\in A⟹\frac{f}{h}$ continuous at $c$

5.2.2 Theorem: Algebraic operations on continuous function on A

Let

• $A\subseteq \mathbb{R}$
• $f,g$ continuous on $A\to \mathbb{R}$
• $b\in \mathbb{R}$

Then

• $f+g,f-g,fg,bf$ continuous on $A$
• $h:A\to \mathbb{R}$ continuous at $c$, $h(x)\ne 0,\forall x\in A⟹\frac{f}{h}$ continuous on $A$

5.2.4 Theorem: Continuous absolute function

Let

• $A\subseteq \mathbb{R}$
• $f:A\to \mathbb{R}$
• $|f|$ defined by $|f|(x):=|f(x)|,\forall x\in A$

Then

• $f$ continuous at $c\in A⟹|f|$ continuous at $c$
• $f$ continuous on $A⟹|f|$ continuous on $A$

5.2.5 Theorem: Continuous square root function

Let

• $A\subseteq \mathbb{R}$
• $f:A\to \mathbb{R}$
• $f(x)\ge 0,\forall x\in A$
• $\sqrt{f}$ defined by $(\sqrt{f})(x):=\sqrt{f(x)},\forall x\in A$

Then

• $f$ continuous at $c\in A⟹\sqrt{f}$ continuous at $c$
• $f$ continuous on $A⟹\sqrt{f}$ continuous on $A$

5.2.6 Theorem: Continuous composite at c

Let

• $A,B\in \mathbb{R}$
• $f:A\to \mathbb{R}$
• $g:B\to \mathbb{R}$
• $f(A)\subseteq \mathbb{B}$

If

• $f$ continuous at $c\in A$
• $g$ continuous at $b=f(c)\in B$

Then

• $g\circ f:A\to \mathbb{R}$ continuous at $c$

5.2.7 Theorem: Continuous composite on A

Let

• $A,B\subseteq \mathbb{R}$
• $f$ continuous on $A\to \mathbb{R}$
• $g$ continuous on $B\to \mathbb{R}$

If

• $f(A)\subseteq B$

Then

• $g\circ f:A\to \mathbb{R}$ continuous on $A$

Intuition

Result of $f(A)$ has to be the same as domain of $g$. i.e., $f(A)\subseteq B$. And note that $g$ is continuous on $B$.