# Category:Continuous Functions

This category contains results about Continuous Functions.

### Continuous Complex Function

As the complex plane is a metric space, the same definition of continuity applies to complex functions as to metric spaces.

### Continuous Real Function

### Continuity at a Point

Let $A \subseteq \R$ be any subset of the real numbers.

Let $f: A \to \R$ be a real function.

Let $x \in A$ be a point of $A$.

### Definition by Epsilon-Delta

Then **$f$ is continuous at $x$** if and only if the limit $\displaystyle \lim_{y \mathop \to x} \map f y$ exists and:

- $\displaystyle \lim_{y \mathop \to x} \, \map f y = \map f x$

### Definition by Neighborhood

Then **$f$ is continuous at $x$** if and only if the limit $\displaystyle \lim_{y \mathop \to x} \map f y$ exists and:

- $\displaystyle \lim_{y \mathop \to x} \, \map f y = \map f x$
- for every $\epsilon$-neighborhood $N_\epsilon$ of $\map f x$ in $\R$, there exists a $\delta$-neighborhood $N_\delta$ of $x$ in $A$ such that $\map f x \in N_\epsilon$ whenever $x \in N_\delta$.

### Continuous Everywhere

Let $f: \R \to \R$ be a real function.

Then $f$ is **everywhere continuous** if and only if $f$ is continuous at every point in $\R$.

### Continuity on a Subset of Domain

Let $A \subseteq \R$ be any subset of the real numbers.

Let $f: A \to \R$ be a real function.

Then **$f$ is continuous on $A$** if and only if $f$ is continuous at every point of $A$.

## Subcategories

This category has the following 2 subcategories, out of 2 total.

## Pages in category "Continuous Functions"

The following 56 pages are in this category, out of 56 total.

### C

- Combination Theorem for Continuous Functions
- Combination Theorem for Continuous Functions/Combined Sum Rule
- Combination Theorem for Continuous Functions/Multiple Rule
- Combination Theorem for Continuous Functions/Product Rule
- Combination Theorem for Continuous Functions/Quotient Rule
- Combination Theorem for Continuous Functions/Sum Rule
- Combination Theorem for Continuous Mappings/Topological Ring/Combined Rule
- Combined Sum Rule for Continuous Functions
- Complex Modulus Function is Continuous
- Concave Real Function is Continuous
- Condition for Continuity on Interval
- Constant Function is Continuous
- Constant Function is Continuous/Real Function
- Constant Real Function is Continuous
- Continuity of Root Function
- Continuous Function on Compact Space is Bounded
- Continuous Function on Compact Space is Uniformly Continuous
- Continuous Image of Closed Interval is Closed Interval
- Continuous Image of Connected Space is Connected/Corollary 2
- Continuous Image of Connected Space is Connected/Corollary 3
- Continuous Injection of Interval is Strictly Monotone
- Continuous Real Function is Darboux Integrable
- Continuous Real Function on Closed Interval is Bijective iff Strictly Monotone
- Continuous Replicative Function
- Convex Real Function is Continuous

### E

### I

### L

### P

### R

### U

- Uniformly Continuous Function is Continuous
- Uniformly Convergent Sequence of Continuous Functions Converges to Continuous Function
- Uniformly Convergent Sequence of Continuous Functions Converges to Continuous Function/Corollary
- Uniformly Convergent Series of Continuous Functions Converges to Continuous Function
- Uniformly Convergent Series of Continuous Functions Converges to Continuous Function/Corollary