Q.Discuss the continuity of the following functions:

Concept: Continuity At A Point — A function is continuous at $x = a$ if $\lim_{x \to a} f(x) = f(a)$. Since $\sin x$ and $\cos x$ are continuous everywhere, any algebraic combination (sum, difference, product) of them is also continuous everywhere.

Reasoning:

1. Both $\sin x$ and $\cos x$ are continuous for all real $x$.
2. The sum, difference, and product of two continuous functions are continuous on their common domain.
3. The domain for all three functions is $\mathbb{R}$. …

The core idea: continuity at a point

A function $f$ is continuous at $x = a$ if three things hold: $f(a)$ is defined, $\lim_{x \to a} f(x)$ exists, and the two are equal. But for most functions we meet in school, we don't need to check this from first principles every time. Instead, we use a powerful theorem: if $f$ and $g$ are continuous at $a$, then $f + g$, $f - g$, and $f \cdot g$ are also continuous at $a$. And we already know that $\sin x$ and $\cos x$ are continuous for all real $x$ — their graphs have no breaks, jumps, or holes anywhere.

So the problem reduces to a simple check: are we combining $\sin x$ and $\cos x$ using only operations that preserve continuity? Yes — addition, subtraction, and multiplication all do. That's the entire logical backbone.

Let's walk through each case.

1. $f(x) = \sin x + \cos x$

Both $\sin x$ and $\cos x$ are continuous on $\mathbb{R}$. The sum of two continuous functions is continuous. Therefore $f(x) = \sin x + \cos x$ is continuous for every real $x$.

2. $f(x) = \sin x - \cos x$

Same logic. Subtraction is just addition of the negative: $\sin x + (-\cos x)$. Since $-\cos x$ is continuous (a constant times a continuous function), the difference is continuous everywhere.

So $f(x) = \sin x - \cos x$ is continuous on $\mathbb{R}$.

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Method: Proving a Combination of Standard Functions Is Continuous on Its Entire Domain

Applies when asked to "discuss the continuity" of a function formed by adding, subtracting, or multiplying together functions already known to be continuous everywhere (such as $\sin x$ and $\cos x$), rather than checking continuity at one specific point.

Steps

Step 1: Identify the building-block functions and confirm each is continuous everywhere

State the known fact: $\sin x$ and $\cos x$ (and any polynomial) are continuous for every real $x$ — no exceptions, no restricted domain.

Step 2: Identify which algebraic operation combines them

Addition, subtraction, and multiplication of two continuous functions preserve continuity at every point of the common domain. (Division only preserves it where the denominator is nonzero — a different case, see the reciprocal-trig-function method.)

Step 3: Apply the appropriate closure rule …

Common Mistakes

Mistake 1: Confusing "oscillating" with "discontinuous"

Why it's wrong: $\sin x$ and $\cos x$ (and their sums/products) oscillate between fixed bounds forever, but oscillation is a completely different property from continuity — a function can swing up and down endlessly and still have no breaks, jumps, or holes anywhere. Correct approach: judge continuity only by whether the one-sided limits and the function value agree at each point, never by how "wild" the graph looks.

Mistake 2: Treating multiplication of two functions as needing extra care compared to addition/subtraction …