Continuity on an interval
Continuity on an Interval
Definition:
 Continuous Function on an Interval: A function $f(x)$ is said to be continuous on an interval $[a,b]$ if it is continuous at every point within the interval $[a,b]$.
Example:
Consider the function $f(x)=\sqrt{x}$
defined on the interval $[0,4]$. Let's examine its continuity on this interval.
Step 1: Checking $f(x)$ on the Interval $[0,4]$:

At $x=0$: $f(0)=\sqrt{0}=0$ (Defined)

At $x=4$: $f(4)=\sqrt{4}=2$(Defined)
Step 2: Investigating the Limit at Endpoints:

Lefthand Limit at $x=0$: ${\mathrm{lim}}_{x\to {0}^{+}}f(x)=\sqrt{x}$

, approaching from the right side as $x$ approaches 0.

Righthand Limit at $x=4$: ${\mathrm{lim}}_{x\to {4}^{}}f(x)=\sqrt{x}$

, approaching from the left side as $x$ approaches 4.
Both these limits exist and are finite.
Step 3: Analysis within the Interval $[0,4]$:
 Intermediate Values: For any $x$ between 0 and 4, $f(x)=\sqrt{x}$
 is defined and continuous.
Conclusion:
The function $f(x)=\sqrt{x}$ is continuous on the interval $[0,4]$ as it is defined, and its limits exist and are finite at the endpoints. Moreover, it's continuous for all values within the interval.