Single-Variable Calculus on an Interval

(i) This setting extends The Real Line: Standing Notation and Background for Calculus on Intervals: its conventions and notation are in force.

(ii) Unless stated otherwise, $I$ denotes an interval (The Real Line: Standing Notation and Background for Calculus on Intervals (v)) containing two distinct points, and $f$ denotes a map $f:I\to\mathbb{R}$.

(iii) Continuity at a point, continuity on a subset and uniform continuity are those of the base setting ((i)).

(iv) Derivatives, differentiability at a point and on a subset, and the derivative map are as in Derivative of a Real Function on an Interval (i), Derivative of a Real Function on an Interval (ii), Derivative of a Real Function on an Interval (iii) and Derivative of a Real Function on an Interval (iv), with uniqueness by Uniqueness of the Derivative of a Real Function on an Interval (i). The increment form of the derivative and continuity of differentiable maps are The Increment Form of the Derivative, and Differentiability Implies Continuity (i) and The Increment Form of the Derivative, and Differentiability Implies Continuity (ii); derivatives of constants, the identity, sums, scalar multiples, products, reciprocals, quotients and powers are Derivatives of Constants, the Identity, Sums, Scalar Multiples, Products, Reciprocals, Quotients and Powers (i), Derivatives of Constants, the Identity, Sums, Scalar Multiples, Products, Reciprocals, Quotients and Powers (ii), Derivatives of Constants, the Identity, Sums, Scalar Multiples, Products, Reciprocals, Quotients and Powers (iv), Derivatives of Constants, the Identity, Sums, Scalar Multiples, Products, Reciprocals, Quotients and Powers (v), Derivatives of Constants, the Identity, Sums, Scalar Multiples, Products, Reciprocals, Quotients and Powers (vi), Derivatives of Constants, the Identity, Sums, Scalar Multiples, Products, Reciprocals, Quotients and Powers (vii), Derivatives of Constants, the Identity, Sums, Scalar Multiples, Products, Reciprocals, Quotients and Powers (viii) and Derivatives of Constants, the Identity, Sums, Scalar Multiples, Products, Reciprocals, Quotients and Powers (iii); the chain rule is Chain Rule for Real Functions on Intervals (i); restriction to subintervals is Restriction Stability of Continuity and of the Derivative (i), Restriction Stability of Continuity and of the Derivative (ii) and Restriction Stability of Continuity and of the Derivative (iii); and inverse functions, their monotonicity, continuity and derivatives are Inverse of a Continuous Strictly Increasing Function on an Interval: Continuity and Derivative (i), Inverse of a Continuous Strictly Increasing Function on an Interval: Continuity and Derivative (ii), Inverse of a Continuous Strictly Increasing Function on an Interval: Continuity and Derivative (iii), Inverse of a Continuous Strictly Increasing Function on an Interval: Continuity and Derivative (iv) and Inverse of a Continuous Strictly Increasing Function on an Interval: Continuity and Derivative (v).

(v) The vanishing of the derivative at an interior maximum or minimum is Vanishing of the Derivative at a Local Extremum Strictly Inside an Interval (i) and Vanishing of the Derivative at a Local Extremum Strictly Inside an Interval (ii); Rolle's theorem is Rolle's Theorem on a Closed Real Interval (i); the mean value theorem is Mean Value Theorem on a Closed Real Interval and on Any Interval (i) and Mean Value Theorem on a Closed Real Interval and on Any Interval (ii); and monotonicity, constancy and Lipschitz bounds from the sign and size of the derivative are The Sign and Size of the Derivative: Monotonicity, Constancy and Lipschitz Bounds (i), The Sign and Size of the Derivative: Monotonicity, Constancy and Lipschitz Bounds (ii), The Sign and Size of the Derivative: Monotonicity, Constancy and Lipschitz Bounds (iii), The Sign and Size of the Derivative: Monotonicity, Constancy and Lipschitz Bounds (iv), The Sign and Size of the Derivative: Monotonicity, Constancy and Lipschitz Bounds (v) and The Sign and Size of the Derivative: Monotonicity, Constancy and Lipschitz Bounds (vi).

(vi) Pointwise and uniform convergence of sequences of real-valued maps are as in Pointwise and Uniform Convergence of a Sequence of Real-Valued Maps (i) and Pointwise and Uniform Convergence of a Sequence of Real-Valued Maps (ii). Continuity and uniform continuity of uniform limits are Continuity and Uniform Continuity of a Uniform Limit of Real-Valued Maps (i), Continuity and Uniform Continuity of a Uniform Limit of Real-Valued Maps (ii), Continuity and Uniform Continuity of a Uniform Limit of Real-Valued Maps (iii) and Continuity and Uniform Continuity of a Uniform Limit of Real-Valued Maps (iv); the uniform Cauchy criterion is The Uniform Cauchy Criterion for Sequences of Real-Valued Maps (i) and The Uniform Cauchy Criterion for Sequences of Real-Valued Maps (ii); and the Weierstrass M-test is The Weierstrass M-Test (i), The Weierstrass M-Test (ii) and The Weierstrass M-Test (iii).

(vii) Partitions of a closed interval, the set of partitions and refinements are as in Partitions of a Closed Interval, the Set of Partitions, and Refinements (i), Partitions of a Closed Interval, the Set of Partitions, and Refinements (ii) and Partitions of a Closed Interval, the Set of Partitions, and Refinements (iii), listed as in The Increasing Listing of a Partition: Endpoints, Lengths, Covering, Unions, Inserting a Point, Concatenation and Uniform Partitions (i), with the properties of The Increasing Listing of a Partition: Endpoints, Lengths, Covering, Unions, Inserting a Point, Concatenation and Uniform Partitions (ii), The Increasing Listing of a Partition: Endpoints, Lengths, Covering, Unions, Inserting a Point, Concatenation and Uniform Partitions (iii), The Increasing Listing of a Partition: Endpoints, Lengths, Covering, Unions, Inserting a Point, Concatenation and Uniform Partitions (iv), The Increasing Listing of a Partition: Endpoints, Lengths, Covering, Unions, Inserting a Point, Concatenation and Uniform Partitions (v), The Increasing Listing of a Partition: Endpoints, Lengths, Covering, Unions, Inserting a Point, Concatenation and Uniform Partitions (vi), The Increasing Listing of a Partition: Endpoints, Lengths, Covering, Unions, Inserting a Point, Concatenation and Uniform Partitions (vii), The Increasing Listing of a Partition: Endpoints, Lengths, Covering, Unions, Inserting a Point, Concatenation and Uniform Partitions (viii) and The Increasing Listing of a Partition: Endpoints, Lengths, Covering, Unions, Inserting a Point, Concatenation and Uniform Partitions (ix). Darboux sums are as in Upper and Lower Darboux Sums of a Bounded Function on a Partition (i), Upper and Lower Darboux Sums of a Bounded Function on a Partition (ii) and Upper and Lower Darboux Sums of a Bounded Function on a Partition (iii), with Bounds for Darboux Sums, Monotonicity under Refinement, the Supremum of the Lower Sums and the Infimum of the Upper Sums, and Concatenation of Partitions (i), Bounds for Darboux Sums, Monotonicity under Refinement, the Supremum of the Lower Sums and the Infimum of the Upper Sums, and Concatenation of Partitions (ii), Bounds for Darboux Sums, Monotonicity under Refinement, the Supremum of the Lower Sums and the Infimum of the Upper Sums, and Concatenation of Partitions (iii), Bounds for Darboux Sums, Monotonicity under Refinement, the Supremum of the Lower Sums and the Infimum of the Upper Sums, and Concatenation of Partitions (iv) and Bounds for Darboux Sums, Monotonicity under Refinement, the Supremum of the Lower Sums and the Infimum of the Upper Sums, and Concatenation of Partitions (v), and with Darboux Sums of Sums, Scalar Multiples and Dominated Functions, and the Oscillation Bound (i), Darboux Sums of Sums, Scalar Multiples and Dominated Functions, and the Oscillation Bound (ii), Darboux Sums of Sums, Scalar Multiples and Dominated Functions, and the Oscillation Bound (iii) and Darboux Sums of Sums, Scalar Multiples and Dominated Functions, and the Oscillation Bound (iv). Riemann integrability and the Riemann integral, over subintervals and oriented, are as in The Riemann Integral on a Closed Interval via Lower and Upper Integrals (i), The Riemann Integral on a Closed Interval via Lower and Upper Integrals (ii), The Riemann Integral on a Closed Interval via Lower and Upper Integrals (iii), The Riemann Integral on a Closed Interval via Lower and Upper Integrals (iv), The Riemann Integral on a Closed Interval via Lower and Upper Integrals (v), The Riemann Integral on a Closed Interval via Lower and Upper Integrals (vi), The Riemann Integral on a Closed Interval via Lower and Upper Integrals (vii) and The Riemann Integral on a Closed Interval via Lower and Upper Integrals (viii). Riemann's criterion is Riemann's Criterion for Integrability, and the Integral Lies between Lower and Upper Sums (i) and Riemann's Criterion for Integrability, and the Integral Lies between Lower and Upper Sums (ii); the rules of the integral are Rules for the Riemann Integral: Sums, Scalar Multiples, Monotonicity, Bounds and Constants (i), Rules for the Riemann Integral: Sums, Scalar Multiples, Monotonicity, Bounds and Constants (ii), Rules for the Riemann Integral: Sums, Scalar Multiples, Monotonicity, Bounds and Constants (iii), Rules for the Riemann Integral: Sums, Scalar Multiples, Monotonicity, Bounds and Constants (iv), Rules for the Riemann Integral: Sums, Scalar Multiples, Monotonicity, Bounds and Constants (v) and Rules for the Riemann Integral: Sums, Scalar Multiples, Monotonicity, Bounds and Constants (vi); additivity over subintervals is Additivity of the Riemann Integral over Adjacent Intervals, Subintervals and Oriented Integrals, and Integrals of Restrictions (i), Additivity of the Riemann Integral over Adjacent Intervals, Subintervals and Oriented Integrals, and Integrals of Restrictions (ii), Additivity of the Riemann Integral over Adjacent Intervals, Subintervals and Oriented Integrals, and Integrals of Restrictions (iii) and Additivity of the Riemann Integral over Adjacent Intervals, Subintervals and Oriented Integrals, and Integrals of Restrictions (iv); integrability of continuous maps is A Continuous Function on a Closed Interval is Riemann Integrable (i) and A Continuous Function on a Closed Interval is Riemann Integrable (ii); the fundamental theorem of calculus is Fundamental Theorem of Calculus, Part I: the Indefinite Integral is Lipschitz, Continuous, and Differentiable at Points of Continuity (i), Fundamental Theorem of Calculus, Part I: the Indefinite Integral is Lipschitz, Continuous, and Differentiable at Points of Continuity (ii), Fundamental Theorem of Calculus, Part I: the Indefinite Integral is Lipschitz, Continuous, and Differentiable at Points of Continuity (iii), Fundamental Theorem of Calculus, Part I: the Indefinite Integral is Lipschitz, Continuous, and Differentiable at Points of Continuity (iv) and Fundamental Theorem of Calculus, Part II: Evaluating an Integral by an Antiderivative (i); integration by parts is Integration by Parts on a Closed Interval (i) and Integration by Parts on a Closed Interval (ii), and by substitution Integration by Substitution on a Closed Interval (i), Integration by Substitution on a Closed Interval (ii) and Integration by Substitution on a Closed Interval (iii); and integrals of uniform limits and of uniformly convergent series are Integrals of Uniform Limits and of Uniformly Convergent Series (i) and Integrals of Uniform Limits and of Uniformly Convergent Series (ii).

(viii) The series of The Real Exponential Function (i), with its terms, absolute convergence and sum map as in The Exponential Series Converges Absolutely, and Its Sum Is a Map on ℝ (i), The Exponential Series Converges Absolutely, and Its Sum Is a Map on ℝ (ii) and The Exponential Series Converges Absolutely, and Its Sum Is a Map on ℝ (iii), the exponential function $\exp$ and the number $e$ are as in The Real Exponential Function (ii) and The Real Exponential Function (iii), with the properties of Basic Properties of the Exponential Function (i), Basic Properties of the Exponential Function (ii), Basic Properties of the Exponential Function (iii), Basic Properties of the Exponential Function (iv), Basic Properties of the Exponential Function (v), Basic Properties of the Exponential Function (vi), Basic Properties of the Exponential Function (vii), Basic Properties of the Exponential Function (viii), Basic Properties of the Exponential Function (ix), Basic Properties of the Exponential Function (x) and Basic Properties of the Exponential Function (xi); the Cauchy product of series is The Cauchy Product of Two Absolutely Convergent Series (i) and The Cauchy Product of Two Absolutely Convergent Series (ii); and the natural logarithm $\log$ is as in The Natural Logarithm (i), with its values The Natural Logarithm (ii) and the properties of Properties of the Natural Logarithm: Values at 1 and e, Products, Quotients, Monotonicity, Continuity, Derivative and the Bound log t at most t - 1 (i), Properties of the Natural Logarithm: Values at 1 and e, Products, Quotients, Monotonicity, Continuity, Derivative and the Bound log t at most t - 1 (ii), Properties of the Natural Logarithm: Values at 1 and e, Products, Quotients, Monotonicity, Continuity, Derivative and the Bound log t at most t - 1 (iii), Properties of the Natural Logarithm: Values at 1 and e, Products, Quotients, Monotonicity, Continuity, Derivative and the Bound log t at most t - 1 (iv), Properties of the Natural Logarithm: Values at 1 and e, Products, Quotients, Monotonicity, Continuity, Derivative and the Bound log t at most t - 1 (v), Properties of the Natural Logarithm: Values at 1 and e, Products, Quotients, Monotonicity, Continuity, Derivative and the Bound log t at most t - 1 (vi) and Properties of the Natural Logarithm: Values at 1 and e, Products, Quotients, Monotonicity, Continuity, Derivative and the Bound log t at most t - 1 (vii).