# 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 129 subcategories, out of 129 total.

### A

- Antiperiodic Functions (6 P)

### B

- Baire Functions (3 P)
- Bernoulli's Inequality (7 P)
- Bolzano-Weierstrass Theorem (10 P)

### C

- Cauchy Sequence is Bounded (5 P)
- Chebyshev's Sum Inequality (7 P)
- Concave Real Functions (17 P)
- Continuum Property (10 P)

### D

- Dedekind's Theorem (5 P)
- Dirichlet Functions (3 P)

### E

- Euler-Gompertz Constant (2 P)
- Even Impulse Pair Function (1 P)

### F

- Farey Sequence (2 P)
- Filtering Function (empty)
- Foiaș Constants (empty)

### G

- Gelfond's Constant (2 P)

### H

- Harmonic Series is Divergent (5 P)
- Heine-Borel Theorem (9 P)
- Hölder Mean (8 P)

### I

### J

### L

- Lindelöf's Lemma (3 P)

### M

- Maximum Value of Real Function (empty)
- Minimum Value of Real Function (empty)

### N

### O

- Odd Impulse Pair Function (empty)
- Orthonormal Sets (3 P)
- Oscillation (5 P)

### P

### Q

- Quadratic Irrationals (4 P)

### R

- Rational Sequences (3 P)
- Rectangle Function (1 P)

### S

- Sampling Function (1 P)
- Signum Function (5 P)
- Stationary Points (4 P)
- Stieltjes Functions (2 P)
- Subdivisions (Real Analysis) (empty)
- Sum of Logarithms (10 P)

### T

- Thomae Function (2 P)
- Triangle Function (empty)

### U

- Uncountable Sum as Series (2 P)

### W

- Weierstrass's Theorem (4 P)

### Y

## Pages in category "Real Analysis"

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

### A

### B

### C

- Carathéodory's Theorem (Analysis)
- Cauchy Sequence is Bounded/Real Numbers
- Cauchy's Convergence Criterion for Series
- Cauchy's Convergence Criterion/Real Numbers
- Characterization of N-Cube
- Chebyshev's Sum Inequality
- Closed Subset of Real Numbers with Lower Bound contains Infimum
- Combination Theorem for Sequences/Real
- Condition for Infimum of Subset to equal Infimum of Set
- Condition for Supremum of Subset to equal Supremum of Set
- Constant Function is Uniformly Continuous/Real Function
- Continuous Extension from Dense Subset
- 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
- Convergence of Taylor Series of Function Analytic on Disk/Lemma/Proof 1
- Convergence of Taylor Series of Function Analytic on Disk/Lemma/Proof 2
- Convergent Generalized Sum of Positive Reals has Countably Many Non-Zero Terms
- 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

### H

### I

- Implicit Function Theorem for Real Functions
- Increasing Real Function has Countable Discontinuities
- 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
- Irrationals are Everywhere Dense in Reals/Normed Vector Space
- Irrationals are Everywhere Dense in Reals/Topology

### L

- Lagrange Polynomial Approximation
- Limit of Bounded Convergent Sequence is Bounded
- Limit of Function by Convergent Sequences/Corollary
- Limit of Image of Sequence/Real Number Line
- Limit of Intersection of Closed Intervals from Zero to Positive Integer Reciprocal
- Lindelöf's Lemma
- Lower Bound of Natural Logarithm
- Lower Sum of Refinement

### M

- Mapping is Bounded on Union iff Bounded on Each Component/Real-Valued Function
- Max Operation Representation on Real Numbers
- Mediant is Between
- Midpoint-Convex Function is Rational Convex
- Min Operation Representation on Real Numbers
- Monotone Convergence Theorem (Real Analysis)
- Monotone Real Function with Everywhere Dense Image is Continuous
- Monotone Real Function with Everywhere Dense Image is Continuous/Lemma
- Moore-Osgood Theorem
- Multiple of Infimum
- Multiple of Supremum

### N

- Negative of Infimum is Supremum of Negatives
- Negative of Lower Bound of Set of Real Numbers is Upper Bound of Negatives
- Negative of Supremum is Infimum of Negatives
- Negative of Upper Bound of Set of Real Numbers is Lower Bound of Negatives
- Number of Type Rational r plus s Root 2 is Irrational
- Number to Reciprocal Power is Decreasing

### O

- Open Cover of Closed and Bounded Real Interval has Finite Subcover
- Open Cover of Closed and Bounded Real Set has Finite Subcover
- Order is Preserved on Positive Reals by Squaring
- Ordering of Series of Ordered Sequences
- Ordering of Squares in Reals
- Oscillation at Point (Infimum) equals Oscillation at Point (Limit)
- Oscillation on Set is an Extended Real Number
- Oscillation on Subset

### P

- Piecewise Continuously Differentiable Function/Definition 2 is Continuous
- Pointwise Addition on Continuous Real Functions forms Group
- Pointwise Limit of Increasing Functions is Increasing
- Positive Real Numbers whose Reciprocals Sum to 1
- Power Function on Base Greater than One is Strictly Increasing/Real Number
- Power Function on Strictly Positive Base is Convex
- Product of Even and Odd Functions
- Product of Increasing Positive Functions is Increasing

### R

- Rationals are Everywhere Dense in Reals
- Rationals are Everywhere Dense in Reals/Normed Vector Space
- Rationals are Everywhere Dense in Reals/Topology
- Real Bounded Monotone Sequence is Convergent
- Real Bounded Sequence has Convergent Subsequence
- Real Cauchy Sequence is Bounded
- 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 Number Line is Banach Space
- Real Numbers Between Epsilons
- Real Plus Epsilon
- Real Power Function on Base Greater than One is Strictly Increasing
- Real Power of Strictly Positive Real Number is Strictly Positive
- Real Rational Function is Continuous
- 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 Real Numbers is Bounded Above iff Set of Negatives is Bounded Below
- Set of Real Numbers is Bounded Below iff Set of Negatives is Bounded Above
- Set of Real Numbers is Bounded Below iff Set of Negatives is Bounded Above/Corollary
- 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
- Span of Exponential of a x by Cosine of b x and Exponential of a x by Sine of b x is preserved under Differentiation wrt x
- 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 and Infima of Combined Bounded Functions
- Suprema of two Real Sets
- Supremum of Absolute Value of Difference equals Difference between Supremum and Infimum
- 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 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