Continuity and Differentiability Short Notes for GATE DA: Concepts, Formulas, Worked Examples & Practice

Continuity and Differentiability short notes for GATE DA: 3 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practice questions.

Chapter Roadmap: Continuity and Differentiability

Chapter Journey

1. Differentiability Rules & Operations

Algebra of derivatives: Sum, Product, Quotient, Chain Rule. Composition logic.

2. Continuity & Functional Constraints

Lipschitz conditions, piecewise continuity, necessary vs sufficient conditions.

Why this matters:
Before solving complex optimization problems, you must know if a function can be differentiated and how to compute it efficiently when functions are combined. This topic provides the toolkit.

The Algebra of Derivatives

Let $f$ and $g$ be differentiable at $x = c$ . The following operations preserve differentiability:

Sum / Difference

(f \pm g)^{'} (c) = f^{'} (c) \pm g^{'} (c)

Product Rule

(f \cdot g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)

Quotient Rule

(\frac{f}{g})^{'} (c) = \frac{f ^{'} ( c ) g ( c ) - f ( c ) g ^{'} ( c )}{[ g ( c ) ] ^{2}}

Condition:

g (c) \neq = 0

Key Insight: The set of differentiable functions forms an algebra. You can build complex differentiable functions from simple ones (polynomials, exponentials, trig) using these operations without returning to first principles.

The Continuity & Differentiability Checklist

Final Exam Checklist

Defined? Is $f (c)$ actually defined?
Limit Exists? Does $LHL = RHL$ ?
Continuous? Does $lim_{x \to c} f (x) = f (c)$ ?
Differentiable? Does $LHD = RHD$ ?
Inequality Check: If $p > 1 ⟹ f^{'} (x) = 0$ .
Abs Value: If $f (c) = 0$ , diff iff $f^{'} (c) = 0$ .

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Continuity and Differentiability Short Notes for GATE DA: Concepts, Formulas, Worked Examples & Practice

Continuity and Differentiability short notes for GATE DA: 3 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practice questions.

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