# Category:Continuous Functions

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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

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

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

### 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 only the following subcategory.

### C

## Pages in category "Continuous Functions"

The following 23 pages are in this category, out of 23 total.

### C

- Complex Modulus Function is Continuous
- Condition for Continuity on Interval
- Constant Function is Continuous
- Continuity of Root Function
- Continuous Function on Compact Space is Bounded
- Continuous Function on Compact Space is Uniformly Continuous
- Continuous Function with Sequential Limits at Infinity has Limit at Infinity
- 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 Replicative Function

### 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/Corollary