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1. Definition of Continuity

A function f(x) is continuous at a point x = a if and only if:

lim (x→a⁻) f(x) = lim (x→a⁺) f(x) = f(a)
Concept: For a function to be continuous, it must be defined at the point, and the limit as it approaches that point from both sides must equal the function's value.

2. Limit Evaluation

Left-Hand Limit (LHL): Approaching from values smaller than a.

lim (x→a⁻) f(x)

Right-Hand Limit (RHL): Approaching from values larger than a.

lim (x→a⁺) f(x)

3. Discontinuity

A function is discontinuous at x = a if:

LHL ≠ RHL
LHL = RHL ≠ f(a)
f(a) is undefined