Question

Discuss the continuity of the function where $f$, where $f$ is defined by
$f(x)=\left\{ \begin{align}
  & 3,\text{ }if\text{ }0\le x\le 1 \\
 & 4,\text{ }if\text{ 1}x<3 \\
 & 5,\text{ }if\text{ 3}\le x\le 10\text{ } \\
\end{align} \right.\text{ }$

Answer 1

Hint: Here we have to apply the condition for continuity that, if Left Hand Limit is equal to the Right Hand Limit which is the same as the function. i.e. $\underset{x\to {{a}^{+}}}{\mathop{\lim }}\,f(x)=\underset{x\to {{a}^{-}}}{\mathop{\lim }}\,f(x)=f(a)$, then the function is continuous. Also check at particular points where the function splits.

Complete step-by-step solution-
Here, consider the function $f$ defined by:
$f(x)=\left\{ \begin{align}
  & 3,\text{ }if\text{ }0\le x\le 1 \\
 & 4,\text{ }if\text{ 1}x<3 \\
 & 5,\text{ }if\text{ 3}\le x\le 10\text{ } \\
\end{align} \right.\text{ }$
So, here we have to check the continuity of the function $f(x)$
By definition we know that a function is said to be continuous in an open interval (a, b) if it is continuous at every point in the interval. For a closed interval, [a, b], f is continuous in (a, b), and
$\underset{x\to {{a}^{+}}}{\mathop{\lim }}\,f(x)=f(a)\text{, }\underset{x\to {{b}^{-}}}{\mathop{\lim }}\,f(x)=f(b)\text{ }$
Similarly, a function is said to be discontinuous at a point $x=a$ if
$\underset{x\to {{a}^{+}}}{\mathop{\lim }}\,f(x)$ and $\underset{x\to {{a}^{-}}}{\mathop{\lim }}\,f(x)$exists but are not equal.
$\underset{x\to {{a}^{+}}}{\mathop{\lim }}\,f(x)$ and $\underset{x\to {{a}^{-}}}{\mathop{\lim }}\,f(x)$exists and are equal but not equal to $f(a)$.
$f(a)$ not defined.
We can solve this in different cases .First consider,
Case 1: Consider the interval $0\le x\le 1$ . Here, the function value is $f(x)=3$, which is a constant function. Every constant function is continuous so we can say that
$f(x)$ is continuous in the interval $0\le x\le 1$.

Case 2: Let us consider the interval $\text{1}x<3$. Here, the function value is $f(x)=4$, which is also a constant function. Therefore $f(x)$ is continuous in the interval $\text{1}x<3$.

Case 3: Here, consider the interval $\text{3}\le x\le 10$. In this interval the function value is $f(x)=5$ which is again a constant function. Hence, $f(x)$ is continuous in the interval $\text{3}\le x\le 10$.

Case 4: Now, let us consider at $x=1$.
We have the Left Hand Limit, $\underset{x\to {{1}^{-}}}{\mathop{\lim }}\,f(x)=3$ and the Right Hand Limit, $\underset{x\to {{1}^{+}}}{\mathop{\lim }}\,f(x)=4$. So we can say that the Left Hand Limit is not equal to the right Hand Limit. i.e.
$\underset{x\to {{1}^{-}}}{\mathop{\lim }}\,f(x)\ne \underset{x\to {{1}^{+}}}{\mathop{\lim }}\,f(x)$
Hence, $f(x)$ is discontinuous at $x=1$.

Case 5: Next, let us consider at $x=3$.
Here, we have the Left Hand Limit, $\underset{x\to {{3}^{-}}}{\mathop{\lim }}\,f(x)=4$ and the Right Hand Limit $\underset{x\to {{3}^{+}}}{\mathop{\lim }}\,f(x)=5$. Therefore, the Left Hand Limit is not equal to the right Hand Limit. i.e.
$\underset{x\to {{3}^{-}}}{\mathop{\lim }}\,f(x)\ne \underset{x\to {{3}^{+}}}{\mathop{\lim }}\,f(x)$.
Hence, $f(x)$ is discontinuous at $x=3$.

The above cases describe the continuity of the function $f(x)$.

Note: In this type of problems, we have to check separately for the points where the function splits. The function will be discontinuous at such particular points, wherever the graph breaks.