Question

Examine the continuity of $f$, where $f$ is defined by
$f\left(x\right)=$ $\{\begin{array}{c}\sin x-\cos x,\text{if}x\ne 0\\ -1,\text{if}x=0\end{array}$

Answer 1

The given function is $f\left(x\right)=\{\begin{array}{c}\sin x-\cos x,\text{if}x\ne 0\\ -1,\text{if}x=0\end{array}$
It is evident that $f$ is defined at all points of the real line.
Let $c$ be a real number.
Case I:
If $c\ne 0$, then $f\left(c\right)=\sin c-\cos c$
$\underset{x\to c}{lim}f\left(x\right)=\underset{x\to c}{lim}(\sin x-\cos x)=\sin c-\cos c$
$\therefore \underset{x\to c}{lim}f\left(x\right)=f\left(c\right)$
Therefore, $f$ is continuous at all points $x$, such that $x\ne 0$
Case II
If $c=0$, then $f\left(0\right)=-1$
$\underset{x\to 0}{lim}f\left(x\right)=\underset{x\to 0}{lim}(\sin x-\cos x)=\sin 0-\cos 0=0-1=-1$
$\underset{x\to 0}{lim}f\left(x\right)=\underset{x\to 0}{lim}(\sin x-\cos x)=\sin 0-\cos 0=0-1=-1$
$\therefore \underset{x\to 0}{lim}f\left(x\right)=\underset{x\to 0}{lim}f\left(x\right)=f\left(0\right)$
Therefore, $f$ is continuous at $x=0$
From the above observations, it can be concluded that f is continuous at every point of the real line.