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

(D) The given function is $f(x) = \begin{cases} 3, & \text{if } 0 \le x \le 1 \\ 4, & \text{if } 1 < x < 3 \\ 5, & \text{if } 3 \le x \le 10 \end{cases}$
To check the continuity at $x=3,$ we evaluate the left-hand limit,right-hand limit,and the value of the function at $x=3.$
$1$. Value of the function at $x=3$:
$f(3) = 5$ (from the third part of the definition).
$2$. Left-hand limit at $x=3$:
$\lim_{x \to 3^-} f(x) = \lim_{x \to 3^-} (4) = 4$.
$3$. Right-hand limit at $x=3$:
$\lim_{x \to 3^+} f(x) = \lim_{x \to 3^+} (5) = 5$.
Since the left-hand limit $(\lim_{x \to 3^-} f(x) = 4)$ is not equal to the right-hand limit $(\lim_{x \to 3^+} f(x) = 5)$,the limit $\lim_{x \to 3} f(x)$ does not exist.
Therefore,the function $f$ is not continuous at $x=3.$