# Continuity on an interval

## Continuity on an Interval

### Definition:

• Continuous Function on an Interval: A function $f\left(x\right)$ is said to be continuous on an interval $\left[a,b\right]$ if it is continuous at every point within the interval $\left[a,b\right]$.

### Example:

Consider the function $f\left(x\right)=\sqrt{x}$

defined on the interval $\left[0,4\right]$. Let's examine its continuity on this interval.

### Step 1: Checking $f\left(x\right)$ on the Interval $\left[0,4\right]$:

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

• At $x=4$: $f\left(4\right)=\sqrt{4}=2$(Defined)

### Step 2: Investigating the Limit at Endpoints:

• Left-hand Limit at $x=0$: ${\mathrm{lim}}_{x\to {0}^{+}}f\left(x\right)=\sqrt{x}$

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

• Right-hand Limit at $x=4$: ${\mathrm{lim}}_{x\to {4}^{-}}f\left(x\right)=\sqrt{x}$

• , approaching from the left side as $x$ approaches 4.

Both these limits exist and are finite.

### Step 3: Analysis within the Interval $\left[0,4\right]$:

• Intermediate Values: For any $x$ between 0 and 4, $f\left(x\right)=\sqrt{x}$
• is defined and continuous.

### Conclusion:

The function $f\left(x\right)=\sqrt{x}$ is continuous on the interval $\left[0,4\right]$ as it is defined, and its limits exist and are finite at the endpoints. Moreover, it's continuous for all values within the interval.
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