Okay, I need to write a professional wiki-style document about continuity calculus. Let me start by the structure. The user provided main category as Education, middle as Mathematics, small as Calculus, and the title Continuity. The document should be in with H1, H2,3 headings, sections like overview, detailed content, references, etc.

First, the title should be H1. Then an overview section explaining what continuity is. Next, I should break down the content into sections. Maybe start with the definition of continuity at a point, then move to continuity over an interval. Then discuss types of discontinuities, properties of continuous functions, and applications. Each main section would be H2, and subsections H3.

For the definition section, I need to explain the three conditions for continuity at a point. Maybe use bullet points here. Then, an example to illustrate that. Examples are good for understanding. Use a simple function like f(x) = x² and check continuity at a point.

Then, continuity over an interval. Closed and open intervals. Mention the difference between continuous on open vs. closed. Maybe another example here.

Types of discontinuities: removable, jump, infinite, oscillating. Each should have a brief explanation and example. For removable, maybe use a function with a hole. Jump could be a piecewise function. Infinite with 1/x². Oscillating like sin(1/x) near 0.

Properties of continuous functions: sum, product, quotient, composition. Also, the Intermediate Value Theorem and Extreme Value Theorem. These are important theorems, so explain them clearly. IVT is useful for proving roots exist.

Applications section could include real-world examples like temperature changes, economic models, and engineering. These show why continuity matters beyond math.

References and related documents at the end. Maybe link to other calculus topics like limits, derivatives, integrals.

I need to make sure the content is accurate. Check the conditions for continuity: limit exists, function defined, limit equals function value. Also, types of discontinuities – removable, jump, infinite, oscillating. Theorems: IVT and EVT. Examples should be simple but illustrative.

Markdown formatting: Use H1 for title, H2 for main sections, H3 for subsections. Bullet points for definitions, examples in code blocks if needed. Maybe use LaTeX for mathematical expressions, like $f(x)$.

Wait, the user mentioned code blocks for technical content. So maybe the mathematical equations should be in code blocks using LaTeX. But in Markdown, LaTeX is usually enclosed in $ or $$, not code blocks. Hmm, but the user said "코드나 기술적 내용의 경우 코드 블록 사용", which translates to using code blocks for code or technical content. But mathematical expressions are usually inline or display. Maybe they meant if there's actual code, like Python for plotting, then use code blocks. So equations can be in LaTeX without code blocks.

Also, ensure the document is at least 800 characters. Let me check the length. Each section adds content, so with examples and explanations, it should be sufficient.

Let me outline the sections:

연속성 (Continuity)

개요

• Continuity is a fundamental concept in calculus, describing functions that have no abrupt changes or jumps.

정의

점에서의 연속성

• Three conditions: f(a) defined, lim x→a f(x) exists, equal to f(a).
• Example: f(x) = x² at x=2.

구간에서의 연속성

• Open interval: continuous at every point.
• Closed interval: also consider endpoints.

불연속의 종류

• Removable (hole)
• Jump (step function)
• Infinite (1/x²)
• Oscillating (sin(1/x))

연속함수의 성질

• Algebra of continuous functions
• Intermediate Value Theorem
• Extreme Value Theorem

응용

• Real-world examples: temperature, economics, engineering.

참고 자료

• Links to related concepts.

Now, flesh out each section with explanations, examples, and theorems. Make sure to explain terms like IVT with their implications. Use bullet points where appropriate. Maybe include a table for types of discontinuities? Or just list them with descriptions.

For the example in point continuity, show step-by-step verification. For IVT, explain that if f(a) and f(b) have opposite signs, there's a root in (a,b).

Check for any inaccuracies. For example, removable discontinuity is when the limit exists but either f(a) is undefined or not equal to the limit. Jump is when left and right limits exist but differ. Infinite is when the limit is ±infinity. Oscillating is when the function doesn't approach any value.

Applications: Continuous functions are essential in modeling real-life scenarios where smooth changes are expected.

Make sure the language is clear and professional, avoiding overly complex sentences. Define terms when first mentioned. Use Korean terms with English equivalents in parentheses if needed.

Now, write the document following this structure, ensuring all sections are covered with sufficient detail and examples.