Discuss the continuity of the function $$f$$, where $$f$$ is defined by $f(x) = \left\{ \begin{array} {1 1} 3 ,& \quad\text{ if  0  $$\leq x$$ $$\leq1$$}\\ 4,& \quad\text{if  1  < $$x$$ < 3 }\\ 5 ,& \quad\text{ if  3  $$\leq x$$ $$\leq10$$}\\ \end{array} \right.$

$\begin{array}{1 1} \text{f is not continuous at} x=-1 \;and \;x=3 \\ \text{f is continuous at}x=1\;and\;x=3 \\ \text{f is not continuous at} x=1 \;and\; x=3 \\ \text{Can not be determined} \end{array}$

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:
In the interval $0\leq x\leq 1$
$f(x)=3$
$f$ is continuous in this interval.
At $x=1$
LHL=$\lim\limits_{\large x\to 1^-}f(x)=3$
RHL=$\lim\limits_{\large x\to 1^+}f(x)=4$
$f$ is discontinuous at $x=1$
Step 2:
At $x=3$
LHL=$\lim\limits_{\large x\to 3^-}f(x)=4$
RHL=$\lim\limits_{\large x\to 3^+}f(x)=5$
LHL $\neq$ RHL.
$f$ is discontinuous at $x=3$
$\Rightarrow f$ is not continuous at $x=1$ and $x=3$