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Continuous functions are fundamental in calculus, representing unbroken, smooth behaviors in graphs. They are defined by three criteria: being defined at a point, having a limit that exists as it approaches the point, and the limit equaling the function's value at that point. Key properties include the Intermediate Value Theorem and the Extreme Value Theorem, which are crucial for understanding function behavior. Continuous functions are also vital in physics, engineering, and economics for modeling and predictions.

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## Definition of Continuous Functions

### Characteristics of Continuous Functions

Continuous functions are unbroken and smooth, allowing for the prediction of function values and essential for the study of limits, derivatives, and integrals

### Criteria for Continuity

Three Criteria for Continuity at a Point

A function must be defined at the point, have a limit as it approaches the point, and have the limit equal to the function's value at that point to be continuous

### Properties of Continuous Functions

Continuous functions have important properties such as the Intermediate Value Theorem, Extreme Value Theorem, and Uniform Continuity, which have significant implications in mathematical analysis

## Applications of Continuous Functions

### Interdisciplinary Applications

Continuous functions are used in fields such as physics, engineering, and economics for modeling systems and processes and making predictions and problem-solving in real-world situations

### Relevance in Calculus

Continuous functions are foundational for the Fundamental Theorem of Calculus, which connects differentiation and integration

### Importance in Computer Science

Continuous functions are crucial for developing algorithms and numerical methods that require smooth and predictable behavior in computer science

## Determining Continuity

### Systematic Approach

To determine continuity, one must ensure the function is defined at the point, compute the limit as the input approaches the point, and verify that the limit matches the function's value at the point

### Strategies for Assessing Continuity

Strategies such as graphing, analyzing the domain, and simplifying expressions can be used to assess continuity, especially around points where the function's behavior may be less obvious

## Real-Life Applications

### Modeling Physical Phenomena

Continuous functions are used to model real-world situations such as the distance traveled over time by a car moving at a constant speed and daily temperature changes

### Importance in Making Accurate Predictions and Analyses

The use of continuous functions in real-life applications highlights their value in both academic studies and professional practice