# Category:Real Analysis

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This category contains results about Real Analysis.

Definitions specific to this category can be found in Definitions/Real Analysis.

**Real analysis** is a branch of mathematics that studies real functions.

## Subcategories

This category has the following 113 subcategories, out of 113 total.

### A

### B

### C

### D

### E

### F

### G

### H

### I

### J

### L

### M

### N

### O

### P

### Q

### R

### S

### T

### U

### W

### Y

## Pages in category "Real Analysis"

The following 182 pages are in this category, out of 182 total.

### B

### C

- Carathéodory's Theorem (Analysis)
- Cauchy Sequence Converges on Real Number Line
- Cauchy Sequence is Bounded/Real Numbers
- Characterization of N-Cube
- Closed Subset of Real Numbers with Lower Bound contains Infimum
- Combination Theorem for Sequences/Real
- Composite of Continuous Mappings is Continuous/Corollary
- Condition for Infimum of Subset to equal Infimum of Set
- Condition for Supremum of Subset to equal Supremum of Set
- Constant Function is Continuous/Real Function
- Constant Function is Uniformly Continuous/Real Function
- Constant Real Function is Continuous
- Continuous Extension from Dense Subset
- Continuous Function on Closed Interval is Uniformly Continuous
- Continuous Image of Closed Interval is Closed Interval
- Continuous Inverse Theorem
- Continuous Midpoint-Concave Function is Concave
- Continuous Midpoint-Convex Function is Convex
- Continuous Real Function is Bounded
- Continuous Strictly Midpoint-Concave Function is Strictly Concave
- Continuous Strictly Midpoint-Convex Function is Strictly Convex
- Continuum Property
- Convergence of Square of Linear Combination of Sequences whose Squares Converge
- Convergence of Taylor Series of Function Analytic on Disk
- Convergence of Taylor Series of Function Analytic on Disk/Lemma
- Convergent Generalized Sum of Positive Reals has Countably Many Non-Zero Terms
- Convergent Real Sequence is Bounded
- Cover Consisting of Open Real Sets has Countable Subcover

### D

- Dedekind's Theorem
- Dedekind's Theorem/Corollary
- Diameter of N-Cube
- Diameter of N-Cube/Corollary
- Dini's Theorem
- Discontinuity of Monotonic Function is Jump Discontinuity
- Distance between Element and Subset of Real Numbers/Examples
- Distance from Subset to Infimum
- Distance from Subset to Supremum
- Distance on Real Numbers is Metric

### E

- Equivalence of Definitions of Distance to Nearest Integer Function
- Equivalence of Definitions of Oscillation at Point for Real Functions
- Equivalence of Definitions of Supremum of Real-Valued Function
- Existence of Sequence in Set of Real Numbers whose Limit is Infimum
- Extreme Value Theorem/Real Function

### H

### I

- Image of Interval by Continuous Function is Interval
- Image of Interval by Continuous Function is Interval/Proof 1
- Image of Interval by Continuous Function is Interval/Proof 2
- Implicit Function Theorem for Real Functions
- Inequality of Hölder Means
- Inequality Rule for Real Sequences
- Infima of two Real Sets
- Infimum of Set of Oscillations on Set
- Infimum of Set of Oscillations on Set is Arbitrarily Close
- Infimum of Subset of Real Numbers is Arbitrarily Close
- Infimum of Upper Sums Never Smaller than Lower Sum
- Infimum Plus Constant
- Infinite Limit Theorem
- Intermediate Value Theorem for Derivatives
- Interval Divided into Subsets
- Irrationals are Everywhere Dense in Reals

### L

- Lagrange Polynomial Approximation
- Limit at Infinity of x^n
- Limit of Bounded Convergent Sequence is Bounded
- Limit of Function by Convergent Sequences/Corollary
- Limit of Function by Convergent Sequences/Real Number Line
- Limit of Image of Sequence/Real Number Line
- Limit of Intersection of Closed Intervals from Zero to Positive Integer Reciprocal
- Limit of Subsequence equals Limit of Real Sequence
- Limit of Subsequence equals Limit of Sequence/Real Numbers
- Lindelöf's Lemma
- Lindelöf's Lemma/Lemma
- Lindelöf's Lemma/Lemma/Lemma
- Lindelöf's Lemma/Lemma/Lemma/Lemma
- Lower Bound of Natural Logarithm

### M

- Mapping Bounded on Union iff Bounded on Each Component/Real-Valued Function
- Maximum Rule for Real Sequences
- Mediant is Between
- Minimum Rule for Real Sequences
- Minkowski's Inequality for Sums/Index 2
- Monotone Convergence Theorem (Real Analysis)
- Monotone Real Function with Everywhere Dense Image is Continuous
- Monotone Real Function with Everywhere Dense Image is Continuous/Lemma
- Monotonicity of Real Sequences
- Moore-Osgood Theorem
- Multiple of Infimum
- Multiple of Supremum

### N

### O

- Odd Power Function is Strictly Increasing/Real Numbers
- Open Cover of Closed and Bounded Real Interval has Finite Subcover
- Open Cover of Closed and Bounded Real Set has Finite Subcover
- Ordering of Squares in Reals
- Oscillation at Point (Infimum) equals Oscillation at Point (Epsilon-Neighborhood)
- Oscillation at Point (Infimum) equals Oscillation at Point (Limit)
- Oscillation at Point (Infimum) equals Oscillation at Point (Limit)/Lemma
- Oscillation on Set is an Extended Real Number
- Oscillation on Subset

### P

### R

- Rationals are Everywhere Dense in Reals
- Real Bounded Monotone Sequence is Convergent
- Real Bounded Sequence has Convergent Subsequence
- Real Cauchy Sequence is Bounded
- Real Convergent Sequence is Cauchy Sequence
- Real Function is Continuous at Point iff Oscillation is Zero
- Real Function is Linearly Dependent with Zero Function
- Real Line Continuity by Inverse of Mapping
- Real Number between Zero and One is Greater than Square
- Real Number Greater than One is Less than Square
- Real Numbers are Close Packed
- Real Numbers Between Epsilons
- Real Plus Epsilon
- Real Power of Strictly Positive Real Number is Strictly Positive
- Real Rational Function is Continuous
- Real Sequence is Cauchy iff Convergent
- Reals are Isomorphic to Dedekind Cuts
- Reciprocal Function is Strictly Decreasing
- Reciprocal Sequence is Strictly Decreasing

### S

- Sequential Continuity is Equivalent to Continuity in the Reals
- Set of Integers is not Bounded
- Set of Local Minimum is Countable
- Set of Strictly Positive Real Numbers has no Smallest Element
- Sign of Odd Power
- Sign of Odd Power/Corollary
- Sign of Quotient of Factors of Difference of Squares
- Sign of Quotient of Factors of Difference of Squares/Corollary
- Square Number Less than One
- Square of Real Number is Non-Negative
- Squeeze Theorem/Sequences/Real Numbers
- Stolz-Cesàro Theorem/Corollary
- Strictly Monotone Real Function is Bijective
- Strictly Positive Integer Power Function is Unbounded Above
- Suprema of two Real Sets
- Supremum of Absolute Value of Difference equals Difference between Supremum and Infimum
- Supremum of Absolute Value of Difference equals Supremum of Difference
- Supremum of Bounded Above Set of Reals is in Closure
- Supremum of Function is less than Supremum of Greater Function
- Supremum of Lower Sums Never Greater than Upper Sum
- Supremum of Set Equals Maximum of Suprema of Subsets
- Supremum of Set Equals Maximum of Supremums of Subsets
- Supremum of Set of Real Numbers is at least Supremum of Subset
- Supremum of Subset of Real Numbers is Arbitrarily Close
- Supremum of Subset of Union Equals Supremum of Union
- Supremum of Sum equals Sum of Suprema
- Supremum Plus Constant
- Surjective Monotone Function is Continuous