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The significance of theorems in calculus lies in their ability to simplify complex mathematical problems and ensure methodological efficiency. Continuity is a key concept, with theorems that dictate the behavior of functions at specific points. These theorems cover operations like sum, difference, product, and quotient, and extend to composite functions, highlighting the predictable nature of continuous functions as opposed to the unpredictability of discontinuous ones.

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## Purpose of Theorems

### Theorems in calculus provide a unified approach to problem-solving and avoid the need for repetitive proofs

## Continuity Theorems

### Definition of Continuity

Continuity is a fundamental concept in calculus that is crucial for understanding the behavior of functions

### Properties of Continuous Functions

The Sum, Difference, Product, Constant Multiple, and Quotient Rules state that operations on continuous functions yield another continuous function, provided certain conditions are met

### Composition of Continuous Functions

The Composition of Continuous Functions theorem ensures that the continuity of individual functions is preserved when they are combined

## Discontinuity

### Unlike continuity, discontinuity does not have a set of general rules due to its unpredictable nature

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