5.1 Continuous Functions

<< 4.3 Some Extensions of the Limit Concept | 5.2 Combinations of Continuous Functions >>

Summary

Definition

• f continuous at c
• f continuous on B

Theorem

Let

• $A\subseteq \mathbb{R}$
• $f:A\to \mathbb{R}$
• $c\in A$

Then

• $f$ continuous at $c$
  • $⟺$ $\forall V_{\epsilon }(f(c)),\exists V_{\delta }(c)\ni x\in V_{\delta }(c)\cap A⟹f(x)\in V_{\delta }(f(c))$
  • $⟺$ $f(x\in V_{\delta }(c)\cap A)\subseteq V_{\epsilon }(f(c))$
  • $⟺$ $\forall (x_n)\in A,(x_n)\to c⟹(f(x_n))\to f(c)$
• $f$ discontinuous at $c$
  • $⟺$ $\exists (x_n)\in A,(x_n)\to c\ni (f(x_n))\nrightarrow f(c)$

---

5.1.1 Definition: Continuous function at a point

Let

• $A\subseteq \mathbb{R}$
• $f:A\to \mathbb{R}$
• $c\in A$

If

• $\forall \epsilon >0,\exists \delta >0\ni |x-c|<\delta ,\forall x\in A⟹|f(x)-f(c)|<\epsilon$

Then

• We say that $f$ is continuous at $c$

Otherwise

• We say that $f$ is discontinuous at $c$

Resembles 4.1.4 Definition: Limit of a function, but swap $L$ with $f(c)$

5.1.5 Definition: Continuous function on a set

Let

• $A\subseteq \mathbb{R}$
• $f:A\to \mathbb{R}$
• $B\subseteq A$

If

• $f(b)$ continuous $\forall b\in B$

Then

• We say that $f$ continuous on $B$

---

5.1.2 Theorem: Neighborhood definition of continuous function

Let

• $A\subseteq \mathbb{R}$
• $f:A\to \mathbb{R}$
• $c\in A$

Then the following statements are equivalent:

• $f$ is continuous at $c$
• $\forall V_{\epsilon }(f(c)),\exists V_{\delta }(c)\ni x\in A\cap V_{\delta }(c)⟹f(x)\in V_{\delta }(f(c))$
• $f(x\in A\cap V_{\delta }(c))\subseteq V_{\epsilon }(f(c))$

Resembles 4.1.6 Theorem: Neighborhood definition of limit of function, but swap $L$ with $f(c)$.

Idfk what the 3rd one means.

5.1.3 Theorem: Sequential criterion

Let

• $A\subseteq \mathbb{R}$
• $f:A\to \mathbb{R}$
• $c\in A$

Then the following statements are equivalent:

• $f$ is continuous at $c$
• $\forall (x_n)\in A,(x_n)\to c⟹(f(x_n))\to f(c)$

Resembles 4.1.8 Theorem: Sequential criterion, but no $x_n\ne c\forall n\in N$ and swap $L$ with $f(c)$

Notice that this implies $f(x)$ has to be defined at $x=c$

5.1.4 Theorem: Discontinuity criterion

Let

• $A\subseteq \mathbb{R}$
• $f:A\to \mathbb{R}$
• $c\in A$

Then the following statements are equivalent:

• $f$ is discontinuous at $c$
• $\exists (x_n)\in A\ni (x_n)\to c,(f(x_n))\nrightarrow f(c)$

Resembles 4.1.9 Theorem: Divergence criteria