Differentiability of a Function on an Interval
Differentiability of a Function on an Interval
Definition:
 Differentiability on an Interval: A function $f(x)$ is considered differentiable on an interval $[a,b]$ if it is differentiable at every point within that interval.
Conditions for Differentiability:

Derivative Existence: The function must have a welldefined derivative at every point within the interval.

Continuity within the Interval: Differentiability implies that the function is also continuous on the interval $[a,b]$.
Example:
Consider the function $f(x)=\sqrt{x}$
defined on the interval $[0,4]$. Let's explore its differentiability on this interval.
Step 1: Derivative of $f(x)=\sqrt{x}$:
The derivative of $f(x)=\sqrt{x}$ is ${f}^{\mathrm{\prime}}(x)=\frac{1}{2\sqrt{x}}$.
Step 2: Checking Differentiability within the Interval $[0,4]$:

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

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

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

is not differentiable at $x=0$ within the interval $[0,4]$ 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 welldefined derivative at every point within the interval for differentiability.

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