Browse Questions

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

Toolbox:
• If $f$ is a real function on a subset of the real numbers and $c$ be a point in the domain of $f$, then $f$ is continuous at $c$ if $\lim\limits_{\large x\to c} f(x) = f(c)$.
Step 1:
LHL=$\lim\limits_{\large x\to 0}(\sin x-\cos x)$
$\quad\;\;=\lim\limits_{\large h\to 0}[\sin (-h)-\cos(-h)]$
$\quad\;\;=\lim\limits_{\large h\to 0}[-\sin h-\cos h]$
$\quad\;\;=-1$
LHL=-1
Step 2:
RHL=$\lim\limits_{\large x\to 0}(\sin x-\cos x)$
$\quad\;\;=\lim\limits_{\large h\to 0}[\sin h-\cos h]$
$\quad\;\;=-1$
RHL=-1
Step 3:
$f(0)=-1$
LHL=RHL=f(0)
$f$ is continuous at $x=0$
$\sin x$ and $\cos x$ both are continuous for all $x\in R$
Therefore $\sin x-\cos x$ is continuous for all $x\in R$
$\Rightarrow f$ is continuous for all $x\in R$.