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:

  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 [a,b].

Example:

Consider the function f(x)=x

defined on the interval [0,4]. Let's explore its differentiability on this interval.

Step 1: Derivative of f(x)=x:

The derivative of f(x)=x is f(x)=12x.

Step 2: Checking Differentiability within the Interval [0,4]:

  • At x=0: The derivative f(x)=12xis undefined at x=0.

  • At x=4: The derivative f(x)=12x=14exists and is finite.

Conclusion:

  • The function f(x)=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 well-defined derivative at every point within the interval for differentiability.

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