Limits, Continuity, and Coercive Functions in Single and Multivariable Calculus

Flashcards providing comprehensive vocabulary coverage of single and multivariable limits, asymptotic comparison symbols, continuity definitions, discontinuity types, key existence theorems (Weierstrass, Bolzano, Darboux, Tonelli), and coercivity properties.

Limit of a Function at a Point ($\varepsilon-\delta$ Definition)

Given a function $f: A \to \mathbb{R}$ with $A \subseteq \mathbb{R}$ and a limit point $x_0$ of $A$, $\lim_{x \to x_0} f(x) = \lambda \in \mathbb{R}$ if for every $\varepsilon > 0$, there exists a $\delta_\varepsilon > 0$ such that for every $x \in A$: $0 < |x - x_0| < \delta_\varepsilon \implies |f(x) - \lambda| < \varepsilon$.

Topological Neighborhood Definition of Limit (Definition 553)

Let $f: A \to \mathbb{R}$ with $A \subseteq \mathbb{R}$ and let $x_0 \in \mathbb{R}$ be a limit point of $A$. We write $\lim_{x \to x_0} f(x) = \lambda \in \mathbb{R}$ if for every neighborhood $V$ of $\lambda$, there exists a neighborhood $U$ of $x_0$ such that $f((U \cap A) \setminus \{x_0\}) \subseteq V$.

Right Limit (Definition 549)

For a function $f: A \to \mathbb{R}$ with $A \subseteq \mathbb{R}$ and a limit point $x_0$ of $A$, $\lim_{x \to x_0^+} f(x) = \lambda \in \mathbb{R}$ if for every $\varepsilon > 0$, there exists $\delta = \delta_\varepsilon > 0$ such that for every $x \in A$, $x_0 < x < x_0 + \delta \implies |f(x) - \lambda| < \varepsilon$.

Left Limit

Denoted by $\lim_{x \to x_0^-} f(x) = \lambda$, it means that for every neighborhood $V_\varepsilon(\lambda)$ of $\lambda$, there exists a left neighborhood of $x_0$ such that if $x \in (x_0 - \delta_\varepsilon, x_0) \cap A$, then $f(x) \in V_\varepsilon(\lambda)$.

Criterion for the Existence of a Two-Sided Limit (Proposition 554)

For $f: A \to \mathbb{R}$ with $A \subseteq \mathbb{R}$ and a point $x_0 \in \mathbb{R}$ where $U(x_0) \setminus \{x_0\} \subseteq A$, $\lim_{x \to x_0} f(x) = \lambda \in \mathbb{R}$ if and only if both one-sided limits exist and are equal: $\lim_{x \to x_0^+} f(x) = \lim_{x \to x_0^-} f(x) = \lambda$.

Vertical Asymptote

A straight line $x = c$ that the graph of a function approaches arbitrarily closely, which occurs if at least one of the one-sided limits satisfies $\lim_{x \to c^-} f(x) = \pm\infty$ or $\lim_{x \to c^+} f(x) = \pm\infty$.

Horizontal Asymptote

A straight line $f(x) = \lambda$ that the graph of $f(x)$ approaches arbitrarily closely as the independent variable grows unbounded, which occurs when $\lim_{x \to +\infty} f(x) = \lambda$ or $\lim_{x \to -\infty} f(x) = \lambda$.

Sequential Characterization of Limits (Proposition 562)

Given $f: A \to \mathbb{R}$ with $A \subseteq \mathbb{R}$ and a limit point $x_0 \in \mathbb{R}$, $\lim_{x \to x_0} f(x) = \lambda$ if and only if $x_n \to x_0 \implies f(x_n) \to \lambda$ for every sequence $(x_n)$ in $A$ with $x_n \neq x_0$ for all $n \ge 1$.

Uniqueness of the Limit (Theorem 565)

A property stating that for a function $f: A \to \mathbb{R}$ with $A \subseteq \mathbb{R}$ and a limit point $x_0 \in \mathbb{R}$, there exists at most one unique limit $\lambda \in \mathbb{R}$ such that $\lim_{x \to x_0} f(x) = \lambda$.

Limit of the Sum of Two Functions Theorem

Let $f, g: A \to \mathbb{R}$ with $A \subseteq \mathbb{R}$ be defined in a neighborhood $U(c)$ of $c \in \mathbb{R}$, possibly excluding $c$. If $\lim_{x \to c} f(x) = l$ and $\lim_{x \to c} g(x) = m$ with $l, m \in \mathbb{R}$, then $\lim_{x \to c} (f(x) + g(x)) = l + m$, proved via sequence limits provided it does not produce the indeterminate form $+\infty - \infty$.

Negligible Function ($o$-notation / Definition 578.1)

A function $f$ is negligible with respect to $g$ as $x \to x_0$ (written $f = o(g)$) if $\lim_{x \to x_0} \frac{f(x)}{g(x)} = 0$, where $g(x) \neq 0$ in a punctured neighborhood of $x_0$.

Comparable Functions (Definition 578.2)

A function $f$ is comparable with $g$ as $x \to x_0$ (written $f \asymp g$) if $\lim_{x \to x_0} \frac{f(x)}{g(x)} = k \neq 0$, where $g(x) \neq 0$ in a punctured neighborhood of $x_0$.

Asymptotic Functions (Definition 578.3)

Two functions $f$ and $g$ are asymptotic to one another as $x \to x_0$ (written $f \sim g$) if $\lim_{x \to x_0} \frac{f(x)}{g(x)} = 1$.

Comparison Criteria / Squeeze Theorem (Theorem 567)

Let $f, g, h: A \to \mathbb{R}$ with $A \subseteq \mathbb{R}$ and let $x_0 \in \mathbb{R}$ be a limit point of $A$. If $g(x) \le f(x) \le h(x)$ for all $x \in A$ and $\lim_{x \to x_0} g(x) = \lim_{x \to x_0} h(x) = \lambda \in \mathbb{R}$, then $\lim_{x \to x_0} f(x) = \lambda$.

Continuous Function at a Limit Point (Definition 586)

A function $f: A \to \mathbb{R}$ with $A \subseteq \mathbb{R}$ is continuous at a limit point $x_0 \in A$ if $\lim_{x \to x_0} f(x) = f(x_0)$. (By convention, $f$ is continuous at every isolated point of $A$.)

Eliminable Discontinuity (Hole)

A type of discontinuity at $x_0 \in A$ where $\lim_{x \to x_0} f(x) = \lambda$ exists and is finite, but $\lambda \neq f(x_0)$.

Jump Discontinuity (Discontinuity of 1st Kind)

A discontinuity where both one-sided limits $\lim_{x \to x_0^+} f(x) = \lambda_1$ and $\lim_{x \to x_0^-} f(x) = \lambda_2$ exist and are finite, but they are unequal ($\lambda_1 \neq \lambda_2$).

Discontinuity of the 2nd Kind

A discontinuity at $x_0 \in A$ where at least one of the one-sided limits $\lim_{x \to x_0^+} f(x)$ or $\lim_{x \to x_0^-} f(x)$ is either $\pm\infty$ or does not exist.

Weierstrass' Theorem (Theorem 612 / Theorem 1084)

A continuous function $f: A \to \mathbb{R}$ with $A \subseteq \mathbb{R}^n$ defined on a compact set $K \subseteq A$ attains both a global minimum and a global maximum on $K$; that is, there exist $x_1, x_2 \in K$ such that $f(x_1) = \min_{x \in K} f(x)$ and $f(x_2) = \max_{x \in K} f(x)$.

Bolzano's Theorem (Theorem 606)

Let $f: [a, b] \to \mathbb{R}$ be a continuous function. If $f(a)f(b) \le 0$, then there exists $c \in [a, b]$ such that $f(c) = 0$. Moreover, if $f$ is strictly monotonic, such $c$ is unique.

Darboux's Theorem (Intermediate Value Theorem / Theorem 616)

Let $f: [a, b] \to \mathbb{R}$ be continuous with $m = \min_{[a, b]} f(x)$ and $M = \max_{[a, b]} f(x)$. For any $z$ with $m \le z \le M$, there exists $c \in [a, b]$ such that $f(c) = z$. If $f$ is strictly monotonic, such $c$ is unique.

Market Equilibrium (Definition 609)

A pair $(p, q) \in [a, b] \times \mathbb{R}_+$ of prices and quantities of some good such that demand equals supply: $q = D(p) = S(p)$.

Budget Set

Given a price vector $p \in \mathbb{R}_{++}^n$ and an income $w > 0$, the set $C(p, w) = \{x \in \mathbb{R}_+^n : p \cdot x \le w\}$, representing all consumption bundles $x$ that are affordable at prices $p$ with wealth $w$.

Coercive Function (Definition 1108)

A function $f: A \to \mathbb{R}$ with $A \subseteq \mathbb{R}^n$ is coercive on a subset $C \subseteq A$ if there exists a scalar $k \in \mathbb{R}$ such that the superlevel set $(f \ge k) \cap C = \{x \in C : f(x) \ge k\}$ is non-empty and compact.

Tonelli's Theorem (Theorem 1118)

A function $f: A \to \mathbb{R}$ with $A \subseteq \mathbb{R}^n$ that is coercive and continuous on a subset $C \subseteq A$ admits a global maximizer in $C$, meaning there exists $x_0 \in C$ such that $f(x_0) = \max_{x \in C} f(x)$.

Supercoercive Function (Definition 1120)

A function $f: \mathbb{R}^n \to \mathbb{R}$ such that for every sequence $(x_n) \subseteq \mathbb{R}^n$, if $\lVert x_n \rVert \to +\infty$, then $f(x_n) \to -\infty$.

Characterization of Supercoercive Functions (Proposition 1124)

A function $f: \mathbb{R}^n \to \mathbb{R}$ is supercoercive if and only if all of its upper contour sets (superlevel sets) are bounded.