Single-Variable Calculus on an Interval

This setting fixes the standing objects and vocabulary of single-variable calculus for items that adopt it. It adopts The Real Line: Standing Notation and Background for Calculus in full, introduces no new concept, and carries its background results by reference.

1. The domain. ¶ $I$ denotes an interval in $\mathbb{R}$ containing at least two points, and $f$ denotes a function $f:I\to\mathbb{R}$. The facts about intervals and interior points collected there are used throughout; in particular, a point lying strictly between two points of $I$ is an interior point of $I$, and $[u,v]\subseteq I$ whenever $u,v\in I$ with $u\le v$.

2. Continuity. ¶ Continuity of $f$ at a point of $I$, continuity of $f$ on $I$, and uniform continuity of $f$ on $I$ are as in The Real Line: Standing Notation and Background for Calculus §continuity.

3. Differentiability. ¶ Let $x_0$ be an interior point of $I$. The function $f$ is differentiable at $x_0$ if there is a real number $L$ such that for every $\varepsilon>0$ there is $\delta>0$ for which every $h\in\mathbb{R}$ with $0<|h|<\delta$ and $x_0+h\in I$ satisfies

Such an $L$ is unique by Uniqueness of the Derivative at an Interior Point and is written $f'(x_0)$. We say that $f$ is differentiable on a subset $J\subseteq I$ all of whose points are interior points of $I$ if $f$ is differentiable at every point of $J$. Differentiability at $x_0$ implies continuity at $x_0$ by Differentiability at an Interior Point Implies Continuity There; the sum, constant multiple and product rules are those of Sum, Constant Multiple, and Product Rules for One-Dimensional Derivatives; the chain rule is that of Chain Rule for One-Dimensional Derivatives; the derivatives of powers and of polynomial functions are those of Derivative of a Polynomial Function on the Real Line; and passing to a subinterval changes neither differentiability nor the value of the derivative, by clause 2 of Restriction Stability of Continuity and of the Derivative.

4. The Riemann integral. ¶ Let $a,b\in\mathbb{R}$ with $a<b$ and $[a,b]\subseteq I$. We say that $f$ is Riemann integrable on $[a,b]$ if the restriction $f|_{[a,b]}$ is Riemann integrable in the sense of that definition, and we write $\int_a^b f(t)\,dt$ for its Riemann integral. A function continuous on $[a,b]$ is Riemann integrable there by A Continuous Function on a Closed Interval is Riemann Integrable §integrable, and its restriction to every closed subinterval $[u,v]\subseteq[a,b]$ with $u<v$ is Riemann integrable on $[u,v]$ by A Continuous Function on a Closed Interval is Riemann Integrable §subintervals. The order bounds of clause 2 of Uniform Partitions and Order Bounds for the Riemann Integral and the additivity over adjacent intervals of Additivity of the Riemann Integral on Adjacent Intervals are used freely.

5. Background theorems. ¶ The following standing results are carried by reference: the extreme value theorem Extreme Value Theorem on a Closed Real Interval; the vanishing of the derivative at an interior local maximum or local minimum, Vanishing of the Derivative at an Interior Local Extremum; Rolle's theorem Rolle's Theorem on a Closed Real Interval; the mean value theorem Mean Value Theorem on a Closed Real Interval; and the two parts of the fundamental theorem of calculus, Fundamental Theorem of Calculus, Part I, on a Closed Real Interval and Fundamental Theorem of Calculus, Part II, on a Closed Real Interval.