Let α ∈ R. Prove that a function f: R →...

MathematicsDifferentiability of a FunctionJEE Advanced 2001Moderate

Let $α \in R$ . Prove that a function $f : R \to R$ is differentiable at $α$ if and only if there is a function $g : R \to R$ which is continuous at $α$ and satisfies $f (x) - f (α) = g (x) (x - α)$ for all $x \in R$ .

Visualized Solution (English)

Carathéodory's Criterion for Differentiability

The Theorem: $f$ is differentiable at $α$ $⟺$ $\exists g$ continuous at $α$ such that $f (x) - f (α) = g (x) (x - α)$ .
This is known as Carathéodory's Criterion.
We need to prove both the Forward and Backward directions.

Forward Direction: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } f is Differentiable

Assume $f$ is differentiable at $α$ .
By definition, the limit $f^{'} (α) = lim_{x \to α} \frac{f ( x ) - f ( α )}{x - α}$ exists and is finite.

Defining .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } g (x) for .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } x \neq = α

For $x \neq = α$ , we define $g (x) = \frac{f ( x ) - f ( α )}{x - α}$ .
This ensures the equation $f (x) - f (α) = g (x) (x - α)$ holds for all $x \neq = α$ .

Defining .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } g (α)

To make $g$ continuous at $α$ , we define $g (α) = lim_{x \to α} g (x)$ .
Since $lim_{x \to α} g (x) = lim_{x \to α} \frac{f ( x ) - f ( α )}{x - α} = f^{'} (α)$ , we set:
Definition: $g (α) = f^{'} (α)$ .

Continuity of .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } g at .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } α

By construction, $lim_{x \to α} g (x) = f^{'} (α)$ .
Also, we defined $g (α) = f^{'} (α)$ .
Since $lim_{x \to α} g (x) = g (α)$ , the function $g$ is continuous at $α$ .

Backward Direction: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } g Exists

Assume there exists a function $g$ that is continuous at $α$ and satisfies:
Equation: $f (x) - f (α) = g (x) (x - α)$ for all $x \in R$ .

Calculating the Derivative .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } f^{'} (α)

Consider the limit for differentiability: $lim_{x \to α} \frac{f ( x ) - f ( α )}{x - α}$ .
From our equation, for $x \neq = α$ , $\frac{f ( x ) - f ( α )}{x - α} = g (x)$ .
Thus, $lim_{x \to α} \frac{f ( x ) - f ( α )}{x - α} = lim_{x \to α} g (x)$ .
Since $g$ is continuous at $α$ , $lim_{x \to α} g (x) = g (α)$ .
Therefore, $f^{'} (α)$ exists and $f^{'} (α) = g (α)$ .

Final Conclusion

Key Takeaway: A function is differentiable at a point if and only if it can be factored into a linear term and a continuous slope function.
Mathematical Result: $f^{'} (α) = g (α)$ .
Next Challenge: Can you use this criterion to prove the Product Rule for derivatives? It becomes much easier with this approach!

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Let $α \in R$ . Prove that a function $f : R \to R$ is differentiable at $α$ if and only if there is a function $g : R \to R$ which is continuous at $α$ and satisfies $f (x) - f (α) = g (x) (x - α)$ for all $x \in R$ .

Visualized Solution (English)

Carathéodory's Criterion for Differentiability

Forward Direction: $f$ is Differentiable

Defining $g (x)$ for $x \neq = α$

Defining $g (α)$

Continuity of $g$ at $α$

Backward Direction: $g$ Exists

Calculating the Derivative $f^{'} (α)$

Final Conclusion

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Visualized Solution (English)

Carathéodory's Criterion for Differentiability

Forward Direction: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } f is Differentiable

Defining .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } g(x) for .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } x=α

Defining .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } g(α)

Continuity of .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } g at .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } α

Backward Direction: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } g Exists

Calculating the Derivative .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } f′(α)

Final Conclusion

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