Maths Volume-7 Part II(Continuity and Differentiability)

Synopsis of Continuity and Differentiability - I

Description: Functions' smoothness, limits, derivatives, and their interrelation define continuity and differentiability concepts.

Synopsis of Continuity and Differentiability - II

Description: Differentiability implies continuity, but continuity doesn’t guarantee differentiability; includes derivative applications.

Graphical Transformation

Description: Scaling, shifting, reflecting, and rotating graphs help visualize function behavior and transformations.

Definition of Continuity - I

Description: A function is continuous if its limit, function value, and approach value match.

Definition of Continuity - II

Description: Continuous functions have no abrupt jumps or breaks in their domain.

Definition of Continuity - III

Description: Mathematically, continuity requires at every point.

Definition of Continuity - IV

Description: Discontinuous functions show gaps, asymptotes, or sudden jumps in values.

Continuity at a Point - I

Description: A function is continuous at if limits exist and match .

Continuity at a Point - II

Description: Pointwise continuity ensures no abrupt changes in function values at specific points.

Continuity at a Point - III

Description: Checking one-sided limits confirms continuity by matching left-hand and right-hand limits.

Continuity for Composition of Function - I

Description: For composition continuity, if and are continuous, then is continuous.

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