# **Exercise Questions** All questions in this exercise are listed below. Click on a question to view its solution. --- --- # **Key Concepts** This exercise focuses on the following concepts: - Continuity at a point: $\lim_{x \to c} f(x) = f(c)$ - Types of discontinuities: removable, jump, infinite, and oscillating - Continuity of polynomials and rational functions - Domain restrictions and vertical asymptotes - Continuity of trigonometric functions and special limits - Piecewise functions and continuity at boundary points - Intermediate Value Theorem (IVT) and existence of roots - Solving for unknown parameters to ensure continuity --- # **Important Formulas** Below are the key formulas used in this exercise: **Continuity Condition:** $$\lim_{x \to c} f(x) = f(c)$$ **Piecewise Continuity at Boundary $c$:** $$\lim_{x \to c^-} f(x) = \lim_{x \to c^+} f(x) = f(c)$$ **Special Trigonometric Limit:** $$\lim_{x \to 0} \frac{\sin x}{x} = 1$$ **Intermediate Value Theorem:** If $f$ is continuous on $[a,b]$ and $k$ is between $f(a)$ and $f(b)$, then $\exists c \in (a,b)$ such that $f(c) = k$. --- # **Summary** This exercise develops proficiency in analyzing function continuity across algebraic, transcendental, and piecewise functions. Problems 1-14 emphasize identifying discontinuities by examining domain restrictions, singularities (where denominators vanish), and behavior at critical points—particularly distinguishing between removable discontinuities (holes) and infinite discontinuities (asymptotes). Problems 15-18 require solving for unknown parameters ($m$, $n$) to enforce continuity, demanding careful calculation of one-sided limits and function values at boundary points. Problem 19 applies the Intermediate Value Theorem to prove existence of solutions within intervals, while Problem 20 examines the theoretical construct of a function discontinuous everywhere. Key strategies include algebraic factorization to simplify rational expressions, evaluating limits of indeterminate forms, and systematic verification of the three continuity conditions.