Real Sine Function is Continuous

Theorem
Let $x \in \R$ be a real number.

Let $\sin x$ be the sine of $x$.

Then:
 * $\sin x$ is continuous on $\R$.

Proof
Recall the definition of the sine function:


 * $\displaystyle \sin x = \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n + 1} } {\paren {2 n + 1}!} = x - \frac {x^3} {3!} + \frac {x^5} {5!} - \cdots$

Thus $\sin x$ is expressible in the form of a power series.

From Sine Function is Absolutely Convergent, we have that the interval of convergence of $\sin x$ is the whole of $\R$.

From Power Series is Differentiable on Interval of Convergence, it follows that $\sin x$ is continuous on the whole of $\R$.