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 functionIf $$p(x)$$ & $$q(x)$$ 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 functionIf $$p(x)$$ & $$q(x)$$ 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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