Question

$f\left(x\right)=\{\begin{array}{c}\left[x\right]+[-x]:x\ne 0\\ \lambda :x=0\end{array}$ Find the value of $\lambda$, $f$ is continuous at =$0$?

Answer 1

For f to be continuous at $x=0$

$\underset{x\to 0^{+}}{lim}f\left(x\right)=\underset{x\to 0^{-}}{lim}f\left(x\right)=f\left(0\right)=\lambda$

$\underset{x\to 0^{+}}{lim}f\left(x\right)=\underset{x\to 0}{lim}f(0+x)=\underset{x\to 0}{lim}\left[x{]}^{+}\right[-x]=0+(-1)=-1$

$\underset{x\to 0^{-}}{lim}f\left(x\right)=\underset{x\to 0}{lim}f(-x)=\underset{x\to 0}{lim}[-x]+\left[x\right]=-1$

$\Rightarrow \lambda =-1$.