Proving the Continuity and Differentiability of sin x

Introduction

In the field of calculus, understanding the continuity and differentiability of basic functions like sine is fundamental. This article will delve into the steps and methods used to prove the continuity of the sin x function. Additionally, we will explore how the differentiability of the sine function directly implies its continuity. Let's begin with the basic definitions and proceed step-by-step.

Continuity of sin x

To prove that the function sin x is continuous at any point (c), we will use the ε-δ definition of continuity. According to this definition, a function f(x) is continuous at a point (c) if for every (ε > 0), there exists a (δ > 0) such that whenever (|x - c|

Step-by-Step Proof

Choose a Point: Let (c) be any real number. We want to show that sin x is continuous at (c).

Set Up the Condition: We need to demonstrate that for any (ε > 0), there exists a (δ > 0) such that if (|x - c|

Use the Sine Difference Identity: We can express the difference (|sin x - sin c|) using the sine difference identity:

sin x - sin c 2 left cosleft( frac{x c}{2} right) sinleft( frac{x-c}{2} right) right

Bound the Cosine Term: Since the cosine function is bounded, we have:

cosleft( frac{x c}{2} right) leq 1

Focus on the Sine Term: Therefore, we can bound the expression:

|sin x - sin c| leq 2 left| sinleft( frac{x-c}{2} right) right| right

Use the Sine Bound: For small angles, the sine function can be approximated by its argument: |sin t| leq |t|. Hence:

|sinleft( frac{x-c}{2} right)| leq left| frac{x-c}{2} right|

Combine the Inequalities: Combining the inequalities, we get:

|sin x - sin c| leq 2 left| frac{x-c}{2} right| |x - c|

Choose (δ): To satisfy (|sin x - sin c|

|sin x - sin c|

Conclusion

Since we can make (|sin x - sin c| sin x is continuous at any point (c). Therefore, sin x is continuous for all real numbers (x).

Implications of Differentiability and Continuity

It is well-known in standard real analysis courses that if a function is differentiable at a point, it is continuous at that point. Therefore, since the derivative of sin x is cos x, which exists for all real numbers, we can conclude that sin x is continuous for all (x).

Keywords

Continuity, Differentiability, sin x