Question

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

Answer 1

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

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

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

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

By Algebra of a continuous function

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

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

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

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

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

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

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

By Algebra of continuous function

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

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

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