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Question

Discuss the continuity of the following functions:
(a) $$f(x)=\sin x+\cos x$$
(b) $$f(x)=\sin x-\cos x$$


Solution

(a) $$f(x)=\sin x+\cos x$$

Let $$p(x)=\sin x$$ & $$q(x)=\cos x$$

We know that $$\sin x$$ & $$\cos x$$ both continuous function

$$\Rightarrow p(x)$$ & $$q(x)$$ is continuous at all real number.

By Algebra of a continuous function

If $$p(x)$$ & $$q(x)$$ are continuous for all real numbers.

Then $$f(x)=p(x)+q(x)$$ is continuous for all real numbers.

$$\therefore f(x)=\sin x+\cos x$$ continuous for all real numbers.

(b) $$f(x)=\sin x-\cos x$$

Let $$p(x)=\sin x$$ & $$q(x)=\cos x$$

We know that $$\sin x$$ & $$\cos x$$ both continuous function.

$$\Rightarrow p(x)$$ & $$q(x)$$ is continuous at all real numbers.

By Algebra of continuous function

If $$p(x)$$ & $$q(x)$$ are continuous for all real numbers.

Then $$f(x)=p(x)-q(x)$$ is continuous for all real numbers.

$$\therefore f(x)=\sin x-\cos x$$ is continuous for all real numbers.

Mathematics

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