Seshu cheera

 Here’s the comprehensive guide for Chapter 5: Continuity & Differentiability (CBSE Class 12 Maths, 2025–26 syllabus): A. 🔑 Key Concepts and Theorems 1. Continuity A function f f is continuous at x = a x = a if lim ⁡ x → a f ( x ) = f ( a ) \lim_{x \to a}f(x) = f(a) , meaning left and right limits equal the function value . Discontinuities : Removable (hole), jump , and infinite/oscillatory . Algebra of continuous functions : Sums, products, quotients (with non-zero denominator), and compositions of continuous functions are continuous . Intermediate Value Theorem / Rolle’s & Mean Value Theorem : continuous on [a,b] ⇒ there exist points satisfying f(b)=f(a) or f′(c) = (f(b)–f(a))/(b–a) . 2. Differentiability f f is differentiable at a a if the limit lim ⁡ h → 0 f ( a + h ) − f ( a ) h \lim_{h \to 0} \frac{f(a+h) - f(a)}{h} exists . Differentiable ⇒ Continuous , but not vice versa (e.g. f ( x ) = ∣ x ∣ f(x)=|x| at 0) . Left ...