# Differentiability of a Function on an Interval

## Differentiability of a Function on an Interval

### Definition:

• Differentiability on an Interval: A function $f\left(x\right)$ is considered differentiable on an interval $\left[a,b\right]$ if it is differentiable at every point within that interval.

### Conditions for Differentiability:

1. Derivative Existence: The function must have a well-defined derivative at every point within the interval.

2. Continuity within the Interval: Differentiability implies that the function is also continuous on 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 explore its differentiability on this interval.

### Step 1: Derivative of $f\left(x\right)=\sqrt{x}$:

The derivative of $f\left(x\right)=\sqrt{x}$ is ${f}^{\mathrm{\prime }}\left(x\right)=\frac{1}{2\sqrt{x}}$.

### Step 2: Checking Differentiability within the Interval $\left[0,4\right]$:

• At $x=0$: The derivative ${f}^{\mathrm{\prime }}\left(x\right)=\frac{1}{2\sqrt{x}}$is undefined at $x=0$.

• At $x=4$: The derivative ${f}^{\mathrm{\prime }}\left(x\right)=\frac{1}{2\sqrt{x}}=\frac{1}{4}$exists and is finite.

### Conclusion:

• The function $f\left(x\right)=\sqrt{x}$

• is not differentiable at $x=0$ within the interval $\left[0,4\right]$ due to the undefined derivative.

• However, it is differentiable for $x>0$ within the interval.

### Understanding Differentiability on an Interval:

• Derivative Existence: A function needs a well-defined derivative at every point within the interval for differentiability.

• Discontinuities Impacting Differentiability: Points of discontinuity often affect differentiability within an interval.