Discuss the continuity of the following function: f(x) = sin(x) + cos(x).

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Let's analyze the continuity of the function $f(x) = \sin(x) + \cos(x)$.

First, we express it in a simplified form: $$ f(x) = \sin x + \cos x $$

We can use a trigonometric identity to reframe this: $$ f(x) = \sqrt{2} \sin\left( x + \frac{\pi}{4} \right) $$

To confirm the continuity at any point $x = a$, where $a \in \mathbb{R}$, we analyze the left-hand limit (LHL) and right-hand limit (RHL) as follows:

1. Left-Hand Limit (LHL): $$ \lim_{x \to a^{-}} f(x) = \lim_{x \to a^{-}} \sqrt{2} \sin\left( x + \frac{\pi}{4} \right) = \sqrt{2} \sin\left( a + \frac{\pi}{4} \right) $$

2. Right-Hand Limit (RHL): $$ \lim_{x \to a^{+}} f(x) = \lim_{x \to a^{+}} \sqrt{2} \sin\left( x + \frac{\pi}{4} \right) = \sqrt{2} \sin\left( a + \frac{\pi}{4} \right) $$

The function value at $x = a$ is $$ f(a) = \sqrt{2} \sin\left( a + \frac{\pi}{4} \right) $$

Thus, for the function to be continuous at each point $a$: $$ LHL = RHL = f(a) $$

Since $LHL$, $RHL$, and $f(a)$ all equal $\sqrt{2} \sin\left( a + \frac{\pi}{4} \right)$ for any $a \in \mathbb{R}$, we conclude that:

The function $f(x) = \sin(x) + \cos(x)$ is continuous at all points.

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