1. Problem: Identify and explain the different types of discontinuity and give clear examples. 2. Key formula and rule: A function $f$ is continuous at $a$ iff $\lim_{x\to a} f(x)=f(a)$. 3. Important rules: Removable discontinuity means the limit exists but $f(a)$ is undefined or not equal to the limit. 4. Important rules: Jump discontinuity means one-sided limits exist but are unequal. 5. Important rules: Infinite discontinuity (vertical asymptote) means one-sided limits are infinite. 6. Important rules: Oscillatory discontinuity means the limit does not exist because the function oscillates arbitrarily near the point. 7. Example (removable): Consider $f(x)=\begin{cases}\frac{x^2-1}{x-1}&x\neq1\\\text{undefined}&x=1\end{cases}$. 8. Simplify the fraction to show the hole; display the cancellation explicitly: $$\frac{x^2-1}{x-1}=\frac{\cancel{(x-1)}(x+1)}{\cancel{x-1}}=x+1$$ 9. Therefore $\lim_{x\to1}\frac{x^2-1}{x-1}=2$ but $f(1)$ is undefined, so there is a removable discontinuity (a hole) at $x=1$. 10. If we define $f(1)=2$ then the function becomes continuous at $x=1$ because $\lim_{x\to1}f(x)=f(1)$. 11. Example (jump): Let $f(x)=\begin{cases}1&x<0\\0&x\ge0\end{cases}$. 12. Compute one-sided limits: $\lim_{x\to0^-}f(x)=1$. 13. Compute one-sided limits: $\lim_{x\to0^+}f(x)=0$. 14. Since the one-sided limits are finite but unequal, there is a jump discontinuity at $x=0$. 15. Example (infinite): Consider $f(x)=\frac{1}{x}$. 16. As $x\to0^+$ we have $\lim_{x\to0^+}\frac{1}{x}=+\infty$ and as $x\to0^-$ we have $\lim_{x\to0^-}\frac{1}{x}=-\infty$. 17. Thus $f$ has an infinite discontinuity and a vertical asymptote at $x=0$. 18. Example (oscillatory): $f(x)=\sin\left(\frac{1}{x}\right)$ for $x\neq0$. 19. $\lim_{x\to0}\sin\left(\frac{1}{x}\right)$ does not exist because the function oscillates between -1 and 1 infinitely often as $x\to0$. 20. How to identify discontinuities in practice: Check points where the function is undefined, compute one-sided limits, attempt algebraic simplification showing cancellation when appropriate (use $\cancel{...}$ to indicate canceled factors), and classify by the behavior of those limits. 21. Final answer: The main types are removable (hole) — example removable at $x=1$; jump (finite different one-sided limits) — example jump at $x=0$; infinite (vertical asymptote) — example infinite at $x=0$; oscillatory (wild nonexistence) — example oscillatory at $x=0$.