Question

Examine the continuity of f, where f is defined by
f(x)=\left\\{\begin{matrix} sin\,x-cos\,x, &if\,x\neq0 \\\ -1,& if\,x=0 \end{matrix}\right.

Answer 1

f(x)=\left\\{\begin{matrix} sin\,x-cos\,x, &if\,x\neq0 \\\ -1,& if\,x=0 \end{matrix}\right.

It is evident that f is defined at all points of the real line.
Let c be a real number.

Case I:
If c≠0,then f(c)=sin c-cos c
$\lim_{x\rightarrow c}f(x)$=$\lim_{x\rightarrow c}$ (sinx-cosx)=sin c-cos c
∴$\lim_{x\rightarrow c}$ f(x)=f(c)
Therefore,f is continuous at all points x,such that x≠0

Case II:
If c=0,then f(0)=-1 and $\lim_{x\rightarrow 0^-}$f(x)=$\lim_{x\rightarrow 0}$(sinx-cosx)=sin0-cos0=0-1=-1
$\lim_{x\rightarrow 0^+}$f(x)=$\lim_{x\rightarrow 0}$(sinx-cosx)=sin0-cos0=0-1=-1
∴$\lim_{x\rightarrow 0^-}$ f(x)=f(0)=$\lim_{x\rightarrow 0^+}$f(x)=f(0)
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.
Thus,f is a continuous function.