Understanding Discontinuities with Formal Definition Examples

📚 Quick Study Guide

🔍 A discontinuity occurs when a function is not continuous at a particular point. This means there's a break, jump, or hole in the graph at that point.

🔢 Formal Definition: A function $f(x)$ is discontinuous at $x=c$ if any of these conditions are met:
• 🚫 $f(c)$ is not defined.
• 📈 $\lim_{x \to c} f(x)$ does not exist.
• ≠ $\lim_{x \to c} f(x) \neq f(c)$.

💡 Types of Discontinuities:
• ✂️ Removable Discontinuity: A hole in the graph, where the limit exists but doesn't equal the function's value at that point.
• 🦘 Jump Discontinuity: The function 'jumps' from one value to another, so the left and right limits exist but are not equal.
• 💥 Infinite Discontinuity: The function approaches infinity (or negative infinity) as $x$ approaches $c$. This often occurs at vertical asymptotes.

Practice Quiz

1. What condition MUST be met for a function $f(x)$ to be continuous at $x=c$?
1. A) $f(c)$ must be defined.
2. B) $\lim_{x \to c} f(x)$ must exist.
3. C) $\lim_{x \to c} f(x) = f(c)$.
4. D) All of the above.
2. Which type of discontinuity is characterized by a 'hole' in the graph?
1. A) Jump Discontinuity
2. B) Infinite Discontinuity
3. C) Removable Discontinuity
4. D) Essential Discontinuity
3. For the function $f(x) = \frac{x^2 - 4}{x - 2}$, what type of discontinuity exists at $x = 2$?
1. A) Jump Discontinuity
2. B) Infinite Discontinuity
3. C) Removable Discontinuity
4. D) No Discontinuity
4. What happens to the function near an infinite discontinuity?
1. A) It approaches a constant value.
2. B) It approaches zero.
3. C) It approaches infinity (or negative infinity).
4. D) It oscillates between two values.
5. If $\lim_{x \to c^-} f(x) = 3$ and $\lim_{x \to c^+} f(x) = 5$, what type of discontinuity exists at $x = c$?
1. A) Removable Discontinuity
2. B) Jump Discontinuity
3. C) Infinite Discontinuity
4. D) No Discontinuity
6. Which of the following functions has a jump discontinuity at $x=0$?
1. A) $f(x) = x^2$
2. B) $f(x) = \frac{1}{x}$
3. C) $f(x) = \begin{cases} 1, & x < 0 \\ 2, & x \geq 0 \end{cases}$
4. D) $f(x) = |x|$
7. For a function to have a removable discontinuity at $x=c$, which of the following must be true?
1. A) The limit exists, and $f(c)$ is defined, but they are not equal.
2. B) The limit does not exist.
3. C) $f(c)$ is not defined.
4. D) The function approaches infinity.