5.3 Continuity, Types of Discontinuity & Asymptotes

Continuity, Types of Discontinuity & Asymptotes

In differential calculus and the UET ECAT entry test, continuity and asymptotic behavior bridge the gap between algebraic function equations and their geometric graphs. ECAT problems frequently require determining whether piecewise branches connect seamlessly, solving for unknown parameters that guarantee continuity, distinguishing between removable holes and infinite vertical asymptotes, and extracting slant asymptotes via polynomial division.

---

1. The Rigorous Definition of Continuity

The Three Core Conditions

A real-valued function $f(x)$ is continuous at a specific point $x = c$ if and only if the following three conditions are met simultaneously:

1. $f(c)$ is defined: The input $c$ lies within the domain of the function: $c \in \text{Dom}(f)$.
2. $\lim_{x \to c} f(x)$ exists: Both one-sided limits exist, are finite, and equal each other: $\lim_{x \to c^-} f(x) = \lim_{x \to c^+} f(x) = L$
3. The limit equals the functional value: $\lim_{x \to c} f(x) = f(c)$

If any of these three conditions fails, the function $f(x)$ is said to be discontinuous at $x = c$.

Continuity on Intervals

• Open Interval $(a, b)$: $f(x)$ is continuous on $(a, b)$ if it is continuous at every single point $c \in (a, b)$.
• Closed Interval $[a, b]$: $f(x)$ is continuous on $[a, b]$ if:
  1. It is continuous on the open interval $(a, b)$.
  2. It is continuous from the right at $a$: $\lim_{x \to a^+} f(x) = f(a)$.
  3. It is continuous from the left at $b$: $\lim_{x \to b^-} f(x) = f(b)$.

Continuity of Standard Function Families

• Polynomials: Continuous everywhere on $(-\infty, \infty)$.
• Rational Functions: Continuous at every point in their domain (discontinuous only where denominator is zero).
• Trigonometric Functions: Continuous across their entire natural domains (e.g., $\tan x$ is continuous on $(-\frac{\pi}{2}, \frac{\pi}{2})$).
• Exponential and Logarithmic Functions: Continuous on their entire natural domains: $e^x$ on $\mathbb{R}$ and $\ln x$ on $(0, \infty)$.
• Hyperbolic Functions: Continuous everywhere on $\mathbb{R}$ ($\sinh x, \cosh x, \tanh x$).

---

2. Foundational Continuity Theorems

1. The Intermediate Value Theorem (IVT)

Let $f$ be continuous on the closed interval $[a, b]$, and let $f(a) \neq f(b)$. If $u$ is any real number strictly between $f(a)$ and $f(b)$, then there exists at least one number $c \in (a, b)$ such that: $f(c) = u$ Geometric Interpretation: A continuous curve cannot pass from a height $y = f(a)$ to a height $y = f(b)$ without crossing every intermediate horizontal line $y = u$.

2. Bolzano's Theorem (Location of Roots)

An immediate and powerful corollary of the IVT: If $f(x)$ is continuous on $[a, b]$ and the endpoints have opposite signs: $f(a) \cdot f(b) < 0$ then there exists at least one real number $c \in (a, b)$ such that $f(c) = 0$. That is, the equation $f(x) = 0$ has at least one real root in $(a, b)$.

3. The Extreme Value Theorem (EVT)

If $f(x)$ is continuous on a closed bounded interval $[a, b]$, then $f$ attains both an absolute maximum $M$ and an absolute minimum $m$ on $[a, b]$.

---

3. Comprehensive Taxonomy of Discontinuities

Discontinuities are classified based on the nature of the failure of the limit or functional value.

1. Removable Discontinuity (Point Hole)

• Definition: The two-sided limit $\lim_{x \to c} f(x) = L$ exists as a finite real number, but either $f(c)$ is undefined or $f(c) \neq L$.
• Remedy: The discontinuity can be "removed" by defining or redefining a single value: $f(c) = L$.
• Geometric Graph: An otherwise smooth curve containing an isolated missing point (a hole).
• Example: $f(x) = \frac{x^2 - 9}{x - 3}$. As $x \to 3$, $\lim_{x \to 3} (x + 3) = 6$, but $f(3)$ is undefined. Setting $f(3) = 6$ restores continuity.

2. Jump (Finite / Step) Discontinuity

• Definition: Both one-sided limits exist as finite numbers, but they are unequal: $\lim_{x \to c^-} f(x) \neq \lim_{x \to c^+} f(x)$
• Jump Magnitude: The vertical jump distance is $|\lim_{x \to c^+} f(x) - \lim_{x \to c^-} f(x)|$.
• Remedy: Cannot be removed by redefining a single point because the two branches approach different elevations.
• Examples:
  • Greatest Integer (Floor) Function: $f(x) = \lfloor x \rfloor$ has jump discontinuities at every integer $k \in \mathbb{Z}$, with jump magnitude $1$.
  • Signum Function: $f(x) = \frac{x}{|x|}$ has a jump of $|1 - (-1)| = 2$ at $x = 0$.

3. Infinite (Essential) Discontinuity

• Definition: One or both one-sided limits approach positive or negative infinity ($+\infty$ or $-\infty$) as $x \to c$.
• Geometric Graph: The curve displays a vertical asymptote at $x = c$.
• Example: $f(x) = \frac{1}{x - 2}$ as $x \to 2$.

4. Oscillating Discontinuity

• Definition: The function oscillates infinitely often within a bounded range as $x \to c$, so no limit exists.
• Example: $f(x) = \sin\left(\frac{1}{x}\right)$ at $x = 0$.

---

4. Piecewise Functions and Parameter Determination

A classic ECAT problem archetype asks candidates to find the numerical values of unknown constants ($k, a, b$) that ensure a piecewise function is continuous across all domain transition boundaries.

General Methodological Protocol

1. Identify the boundary transition points $x = c$.
2. Compute the left-hand limit: $\text{LHL} = \lim_{x \to c^-} f_1(x)$.
3. Compute the right-hand limit: $\text{RHL} = \lim_{x \to c^+} f_2(x)$.
4. Evaluate the assigned functional value $f(c)$.
5. Equate all three quantities: $\text{LHL} = \text{RHL} = f(c)$, and solve the resulting algebraic equation(s).

Multi-Parameter Example

Determine $a$ and $b$ such that $f(x)$ is continuous everywhere on $\mathbb{R}$: $f(x) = \begin{cases} ax + 3, & x < 2 \\ 5, & x = 2 \\ 2x^2 + b, & x > 2 \end{cases}$

• $\lim_{x \to 2^-} f(x) = a(2) + 3 = 2a + 3$.
• $\lim_{x \to 2^+} f(x) = 2(2)^2 + b = 8 + b$.
• For continuity: $2a + 3 = 5 \implies 2a = 2 \implies a = 1$.
• For continuity: $8 + b = 5 \implies b = -3$.
• Result: $a = 1, b = -3$.

---

5. Asymptotes of Rational and Algebraic Functions

An asymptote is a straight line such that the perpendicular distance between the curve $y = f(x)$ and the line tends to zero as the coordinates on the curve tend to infinity.

1. Vertical Asymptotes (VA)

The vertical line $x = a$ is a vertical asymptote of $y = f(x)$ if at least one of the following holds: $\lim_{x \to a^-} f(x) = \pm\infty \qquad \text{or} \qquad \lim_{x \to a^+} f(x) = \pm\infty$

Systematic Procedure for Rational Functions: Given $R(x) = \frac{P(x)}{Q(x)}$:

1. Factor both $P(x)$ and $Q(x)$ completely.
2. Cancel all common binomial factors $(x - r)$ between numerator and denominator to reduce $R(x)$ to lowest terms.
3. The real zeros of the reduced denominator determine the vertical asymptotes: $Q_{\text{reduced}}(x) = 0$.

ECAT Distinction — Holes vs. Vertical Asymptotes: If a factor $(x - c)$ appears in both numerator and denominator with multiplicity $k$ in $P(x)$ and $m$ in $Q(x)$:

• If $k \ge m$: There is a removable point hole at $x = c$.
• If $m > k$: There is a vertical asymptote at $x = c$.

2. Horizontal Asymptotes (HA)

The horizontal line $y = L$ is a horizontal asymptote of $y = f(x)$ if: $\lim_{x \to \infty} f(x) = L \qquad \text{or} \qquad \lim_{x \to -\infty} f(x) = L$

For a rational function $R(x) = \frac{P(x)}{Q(x)} = \frac{a_n x^n + \dots + a_0}{b_m x^m + \dots + b_0}$:

• Case 1: $\text{deg}(P) < \text{deg}(Q)$ ($n < m$): The horizontal asymptote is the x-axis: $y = 0$.
• Case 2: $\text{deg}(P) = \text{deg}(Q)$ ($n = m$): The horizontal asymptote is the ratio of leading coefficients: $y = \frac{a_n}{b_m}$.
• Case 3: $\text{deg}(P) > \text{deg}(Q)$ ($n > m$): There is no horizontal asymptote.

Note on Crossing Asymptotes: While a curve can never intersect a vertical asymptote (as the function is undefined there), a curve can cross a horizontal or slant asymptote at finite values of $x$.

3. Slant (Oblique) Asymptotes

When the degree of the numerator is exactly one greater than the degree of the denominator ($n = m + 1$), the rational function has a slant (oblique) asymptote of the form: $y = mx + c$

Extraction via Polynomial Long Division: Divide $P(x)$ by $Q(x)$: $\frac{P(x)}{Q(x)} = (mx + c) + \frac{R(x)}{Q(x)}$ where the remainder degree is strictly less than $m$. As $x \to \pm\infty$, $\frac{R(x)}{Q(x)} \to 0$. Therefore, the linear quotient $y = mx + c$ is the equation of the slant asymptote.

---

6. Discontinuity and Asymptote Taxonomy Table

---

7. High-Speed ECAT Inspection Heuristics

Heuristic 1: Total Asymptotes of a Rational Curve

To find the total number of asymptotes of $R(x) = \frac{P(x)}{Q(x)}$ in under 20 seconds:

1. Factor denominator to find number of distinct real non-canceling roots $\implies N_V$ vertical asymptotes.
2. Compare degrees of $P$ and $Q$:
   • If $\text{deg}(P) \le \text{deg}(Q)$: Exactly $1$ horizontal asymptote ($N_H = 1, N_S = 0$).
   • If $\text{deg}(P) = \text{deg}(Q) + 1$: Exactly $1$ slant asymptote ($N_H = 0, N_S = 1$).
   • If $\text{deg}(P) > \text{deg}(Q) + 1$: $N_H = 0, N_S = 0$.
3. Total asymptotes = $N_V + N_H + N_S$.