Proof that the Absolute Value Function is Not Differentiable at Zero

TL;DR

Absolute value of x is not differentiable at zero due to differing one-sided limits.

Transcript

we have to prove that the absolute value of x is not differentiable at zero so uh proof so we'll start by looking at the limit as x approaches c of f of x minus f of c over x minus c so this is what we have to use so if a function is differentiable at c then this limit will exist so in our case f of x is the square root of x and c is zero so let's ... Read More

Key Insights

• ☺️ Calculating the limit as x approaches c is vital to determining differentiability.
• 🥺 Differing one-sided limits indicate abrupt changes in the function, leading to non-differentiability.
• 😥 The absolute value function exhibits non-differentiability at specific points like zero.
• ⛔ Understanding one-sided limits is essential for analyzing the behavior of functions.
• 🫥 Non-differentiability suggests the absence of a unique tangent line at a given point.
• 🙈 Continuous functions may not always be differentiable, as seen with piecewise functions like |x|.
• 🖐️ The concept of differentiability plays a crucial role in calculus and function analysis.

Questions & Answers

Q: How do you establish the differentiability of a function at a given point?

The existence of the limit as x approaches c of f(x) minus f(c) over x minus c determines differentiability at c. If the limit exists, the function is differentiable at that point.

Q: Why does differing one-sided limits result in non-differentiability?

Differing one-sided limits signify that the function fails to have a unique tangent line at that point, indicating non-differentiability due to abrupt changes in slope.

Q: What role do one-sided limits play in determining differentiability?

One-sided limits help identify discontinuities and abrupt changes in function behavior, crucial in determining whether a function is differentiable at a specific point.

Q: How does non-differentiability impact the continuity of a function?

Non-differentiability does not necessarily imply non-continuity. A function can be continuous but not differentiable, as seen with the absolute value function f(x)=|x|.

Summary & Key Takeaways

• To prove non-differentiability, analyze the limit as x approaches c of f(x) minus f(c) over x minus c.

• Using the absolute value function f(x)=|x|, show that one-sided limits differ at x=0.

• Conclude that the function f(x)=|x| is not differentiable at c=0, despite being continuous.