Real-World Examples of Jump and Infinite Discontinuities

📚 Quick Study Guide

• 🔍 A discontinuity occurs at a point where a function is not continuous.
• 📈 A jump discontinuity happens when the function "jumps" from one value to another at a specific point. Formally, $\lim_{x \to c^-} f(x) \neq \lim_{x \to c^+} f(x)$, and both limits exist.
• ♾️ An infinite discontinuity occurs when the function approaches infinity (or negative infinity) as $x$ approaches a certain value. This often happens with rational functions where the denominator approaches zero.
• 📝 Key differences: Jump discontinuities have finite (but unequal) left and right limits, while infinite discontinuities have at least one infinite limit.
• 💡 Recognizing them visually: Jump discontinuities look like a step, while infinite discontinuities look like a vertical asymptote.
• 📐 Examples of jump discontinuities: Piecewise functions defined with different values at the boundary.
• ➗ Examples of infinite discontinuities: Functions like $f(x) = \frac{1}{x}$ at $x=0$ or $f(x) = \tan(x)$ at $x = \frac{\pi}{2}$.

🧪 Practice Quiz

1. Which of the following functions has a jump discontinuity at $x = 2$?
   1. $f(x) = \begin{cases} x+1, & x < 2 \\ x^2, & x \geq 2 \end{cases}$
   2. $f(x) = \frac{1}{x-2}$
   3. $f(x) = x^2 - 4$
   4. $f(x) = \sqrt{x-2}$
2. Which of the following functions has an infinite discontinuity at $x = -1$?
   1. $f(x) = \frac{1}{x+1}$
   2. $f(x) = x+1$
   3. $f(x) = \frac{x}{x+2}$
   4. $f(x) = \sqrt{x+1}$
3. Consider the function $f(x) = \begin{cases} 3x, & x < 1 \\ 5, & x \geq 1 \end{cases}$. What type of discontinuity exists at $x = 1$?
   1. Removable Discontinuity
   2. Jump Discontinuity
   3. Infinite Discontinuity
   4. No Discontinuity
4. Which function has an infinite discontinuity at $x=3$?
   1. $f(x) = \frac{x-3}{x+3}$
   2. $f(x) = \frac{1}{x-3}$
   3. $f(x) = x^2 - 9$
   4. $f(x) = \frac{1}{x^2+9}$
5. The function $f(x) = \frac{1}{(x-5)^2}$ has what type of discontinuity at $x = 5$?
   1. Jump Discontinuity
   2. Removable Discontinuity
   3. Infinite Discontinuity
   4. No Discontinuity
6. For what value of $x$ does the function $f(x) = \tan(x)$ have an infinite discontinuity within the interval $[0, \pi]$?
   1. $0$
   2. $\frac{\pi}{4}$
   3. $\frac{\pi}{2}$
   4. $\pi$
7. Which of the following functions demonstrates a jump discontinuity at $x=0$?
   1. $f(x) = |x|$
   2. $f(x) = \begin{cases} -1, & x < 0 \\ 1, & x > 0 \end{cases}$
   3. $f(x) = x^3$
   4. $f(x) = \frac{1}{x}$