Differentiability

I've drafted an initial article on Differentiability. Please review.

Review by Critic Carla:

The article is well-written, technically accurate, and adheres to an encyclopedic tone. It correctly addresses the topic of "Differentiability" and provides a comprehensive overview from basic definitions to advanced applications.

However, to reach a professional Wikipedia standard, I suggest the following improvements:

• Expand the "Notable Contributors" section of the infobox. The infobox currently lists "Not specified," but the "Historical Development" section of the article explicitly mentions Isaac Newton, Gottfried Wilhelm Leibniz, Augustin-Louis Cauchy, and Karl Weierstrass. These should be moved into the infobox for quick reference.
• Add a dedicated section on "Differentiability vs. Continuity." While this is mentioned in the lead, the relationship between these two concepts is a cornerstone of mathematical analysis. A standalone section would allow for a more detailed explanation of why differentiability implies continuity but not vice versa, perhaps including more examples beyond the absolute value function.
• Include a subsection on "Rules of Differentiation." The article explains what differentiability is, but it skips the practical side of how it is applied. Adding a subsection discussing the linearity of the derivative, the product rule, and the chain rule would provide a more complete educational resource.
• Improve the "Formal Definition" formatting. The list of smoothness classes ($C^0, C^1, C^k, C^\infty$) is currently presented as separate unordered lists. These should be consolidated into a single, cohesive bulleted list or a table to improve readability and visual structure.

Fact-check by Fact-Checker Finn:

• Absolute language: The claim that without differentiability, the application of the Fundamental Theorem of Calculus "would be impossible" is an overstatement; the theorem's general versions (like the Lebesgue differentiation theorem) apply to a broader class of functions.
• Vague terminology: The article describes the absolute value function $f(x) = |x|$ at $x=0$ as having a "cusp." In precise mathematical terms, $|x|$ has a "corner" or "kink"; a "cusp" typically refers to a point where the derivative approaches infinity from both sides (e.g., $f(x) = x^{2/3}$).
• Missing nuance: The statement that "every differentiable function is necessarily continuous" is true for the standard definition of the derivative, but the article fails to mention that this relationship is specific to functions of a real variable, lacking nuance regarding the broader context of different types of continuity/differentiability in advanced analysis.
• Sweeping generalization: The assertion that differentiability is applied across "virtually every quantitative science" is promotional and lacks a specific citation to support the scope of the claim.

Review complete. Some issues were flagged. @Scientist Sage, please address the feedback.