Proving differentiability for a function from the definition

In summary, proving differentiability for a function from the definition involves showing that the limit of the difference quotient exists. Specifically, for a function \( f(x) \) to be differentiable at a point \( a \), the limit \( \lim_{h \to 0} \frac{f(a+h) - f(a)}{h} \) must exist and be finite. This limit represents the derivative \( f'(a) \). If the limit does not exist or is infinite, the function is not differentiable at that point. Additionally, differentiability implies continuity, but continuity alone does not guarantee differentiability.
  • #1
member 731016
Homework Statement
Please see below
Relevant Equations
Please see below
For this problem,
1715475811269.png

The solution is,
1715475847832.png

However, does someone please know why we allowed to assume that the derivative exists for f i.e ##f'(a) = \lim_{x \to a} \frac{f(x) - f(a)}{x - a}##?

Thanks!
 
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  • #2
ChiralSuperfields said:
Homework Statement: Please see below
Relevant Equations: Please see below

For this problem,
View attachment 345049
The solution is,
View attachment 345050
However, does someone please know why we allowed to assume that the derivative exists for f i.e ##f'(a) = \lim_{x \to a} \frac{f(x) - f(a)}{x - a}##?

Thanks!
Because of the first sentence: Let ##f## be a differentiable function.
 
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FAQ: Proving differentiability for a function from the definition

What is the definition of differentiability for a function?

Differentiability at a point means that the derivative of the function exists at that point. A function f(x) is said to be differentiable at x = a if the limit of the difference quotient as x approaches a exists. Mathematically, this is expressed as:

f'(a) = lim (h -> 0) [f(a + h) - f(a)] / h.

How do you prove that a function is differentiable at a specific point?

To prove that a function f(x) is differentiable at a specific point x = a, you need to show that the limit of the difference quotient exists as h approaches 0. This involves calculating:

lim (h -> 0) [f(a + h) - f(a)] / h and demonstrating that this limit yields a finite number.

What are the common pitfalls when proving differentiability?

Common pitfalls include failing to correctly evaluate the limit, overlooking the existence of the limit, or assuming differentiability without checking the necessary conditions. It's important to ensure that the limit is not only finite but also that it approaches the same value from both sides of a.

Can a function be continuous at a point but not differentiable there?

Yes, a function can be continuous at a point but not differentiable. A classic example is the absolute value function f(x) = |x| at x = 0. The function is continuous at that point, but the left-hand and right-hand derivatives do not match, meaning the derivative does not exist there.

What role does continuity play in differentiability?

Continuity is a necessary condition for differentiability. If a function is differentiable at a point, it must be continuous at that point. However, the converse is not true; a function can be continuous without being differentiable. Thus, while continuity is required for differentiability, it is not sufficient on its own.

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