Real analysis

• Find the Infimum and Supremum of the Set
  Separate the even and odd cases of n to find the exact bounds: the set has minimum −1 and maximum 2, so its infimum and supremum are attained.
• How to Integrate the Incomplete Beta Function
  An antiderivative of the lower incomplete beta function is x B_x(a,b) − B_x(a+1,b) + C; the integral from 0 to 1 is B(a,b+1).
• When Is the Absolute Value of a Function Continuous?
  If f is continuous, then |f| is continuous. The converse is generally false, with a simple sign-changing counterexample.
• Difference Between Supremum and Upper Bounds
  An upper bound is any number above a set, while its supremum is the least such number; infimum is the greatest lower bound.
• How to Calculate the Factorial of a Noninteger
  The factorial of a noninteger is calculated with the Gamma function, giving values such as 1/2! = √π/2 and 3.5! = 105√π/16.
• How to Find the Support of a Function
  Find a function’s support by taking the closure of the points where it is nonzero; learn how endpoints, domains, and almost-everywhere definitions affect it.
• How to Tell Whether a Function Is Integrable
  On a closed bounded interval, a function is Riemann integrable exactly when it is bounded and its discontinuities have measure zero.
• Examples of Continuous Strictly Increasing Functions on ℝ
  The requested property is missing from the title; arctan is continuous and strictly increasing on ℝ, yet bounded and not onto ℝ.
• Does a Connected Graph Imply That a Function Is Continuous?
  A function can be discontinuous even when its graph is connected. The oscillating function sin(1/x), extended by f(0)=0, is a concrete example.
• What Is an Example of an Unbounded Set?
  The natural numbers are an example of an unbounded set: no finite real number is an upper bound for all of their elements.
• Why Does a Nonempty Bounded Set of Reals Have a Supremum?
  A nonempty set of real numbers that is bounded above has a supremum because the real numbers satisfy the least-upper-bound property.
• Can a Double Sequence Have Different Iterated Limits?
  A double sequence can have different iterated limits: for aₘₙ = m/(m+n), the two orders give 0 and 1, so no joint limit exists.
• Does a Countably Infinite Sum of Zeros Equal Zero?
  A countable series whose every term is zero has sum zero: every partial sum is zero, so its limit is zero, regardless of enumeration.
• What Is the Difference Between an Arbitrary and a Finite Collection?
  An arbitrary collection may have any number of members, while a finite collection has finitely many; see how this affects unions and intersections.
• What Is the Difference Between Supremum and Maximum?
  A supremum is the least upper bound and need not belong to a set; a maximum is the greatest element, so every maximum is a supremum but not vice versa.
• Why Are Constant Functions Measurable?
  A constant function is measurable because every measurable-set preimage is either the whole domain or the empty set, both in any sigma-algebra.
• Find the Supremum, Infimum, lim sup, and lim inf of 2 − 1/n
  For n ≥ 1, the sequence 2 − 1/n has infimum and minimum 1, supremum 2, and both lim sup and lim inf equal to 2.
• Does Every Nonempty Bounded-Above Set of Real Numbers Have a Supremum?
  Every nonempty subset of the real numbers that is bounded above has a supremum, by the completeness property of the real numbers.
• How to Approximate Simple Functions by Step Functions Almost Uniformly
  On a finite interval, approximate each measurable level set by finite unions of intervals; summable errors make the resulting step functions converge almost uniformly.
• Baby Rudin Chapter 4 Exercises 5 and 6: Solutions
  Solutions to Baby Rudin Chapter 4 Exercises 5 and 6: use the intermediate value theorem for constant sign and continuity to prove the zero set is closed.
• A Finitely Additive Measure Is Countably Additive Exactly When It Is Continuous from Below
  A nonnegative finitely additive set function on a sigma-algebra is countably additive if and only if its values converge along every increasing sequence of sets.
• What Is the Difference Between Bounded, Unbounded, Open, and Closed Sets?
  Open, closed, bounded, and unbounded describe different properties. Learn their definitions, classify common subsets of the real line, and avoid common mix-ups.
• How to Calculate the Supremum of a Set
  To calculate a supremum, find an upper bound and prove set elements come arbitrarily close to it; unlike a maximum, the supremum need not belong to the set.
• Are Singleton Sets in ℝ Both Open and Closed?
  In ℝ with the usual topology, a singleton {a} is closed but not open. Proofs of both facts, and why only ∅ and ℝ are both open and closed.
• Is Every Singleton Set Open? (No, Here's When It Is)
  Singleton sets are not open in general. In ℝ, {0} is closed but not open. A singleton {x} is open exactly when x is an isolated point, as in discrete spaces.
• Why Is Every Singleton Set Open in a Finite Metric Space?
  Proof that {x} is open in any finite metric space, using the minimum distance to the other points, and why this makes every subset open (the discrete topology).