determine the points in the interval $(1, 6)$ at which the function has discontinuities. for each point state the conditions in the continuity checklist that are violated. in order for $f$ to be continuous at $a$, the following three conditions must hold.
1. $f(a)$ is defined ($a$ is in the domain of $f$).
2. $\lim_{x \to a} f(x)$ exists.
3. $\lim_{x \to a} f(x) = f(a)$ (the value of $f$ equals the limit of $f$ at $a$).

$f(x)$ is discontinuous at $x = \box$
(use a comma to separate answers as needed.)

Question

determine the points in the interval $(1, 6)$ at which the function has discontinuities. for each point state the conditions in the continuity checklist that are violated. in order for $f$ to be continuous at $a$, the following three conditions must hold.
1. $f(a)$ is defined ($a$ is in the domain of $f$).
2. $\lim_{x \to a} f(x)$ exists.
3. $\lim_{x \to a} f(x) = f(a)$ (the value of $f$ equals the limit of $f$ at $a$).

$f(x)$ is discontinuous at $x = \box$
(use a comma to separate answers as needed.)

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [], "new_concepts": [ "Graphical Continuity Analysis", "Removable Discontinuity", "Jump Discontinuity" ], "current_concepts": [ "Graphical Continuity Analysis", "Removable Discontinuity", "Jump Discontinuity", "Infinite Discontinuity" ] } ## Identify potential discontinuity points in the interval (1, 6) $$ x \in \{2, 3, 5\} $$ ## Evaluate continuity conditions at each candidate point $$ \begin{aligned} &\text{At } x = 2:\ &\quad \lim_{x \to 2} f(x) = \infty \implies \text{Limit does not exist (Infinite discontinuity)}\ &\text{At } x = 3:\ &\quad \lim_{x \to 3^-} f(x) = 2,\quad \lim_{x \to 3^+} f(x) = 1 \implies \lim_{x \to 3} f(x) \text{ does not exist (Jump discontinuity)}\ &\text{At } x = 5:\ &\quad \lim_{x \to 5} f(x) = 3,\quad f(5) = 3 \implies \text{Continuous} \end{aligned} $$ ## List the points of discontinuity in the open interval (1, 6) $$ x = 2, 3 $$ $f(x)$ is discontinuous at $x =$ $2, 3$ { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Calculus", "Graphical Continuity Analysis" ] }

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Question

Explanation:

Identify potential discontinuity points in the interval (1, 6)

Evaluate continuity conditions at each candidate point

List the points of discontinuity in the open interval (1, 6)

Answer: