CalcRef

Limits & Continuity · Continuity & Theorems

Continuity Definition

f is continuous at a  ⟺  lim⁡x→af(x)=f(a)f \text{ is continuous at } a \iff \lim_{x \to a} f(x) = f(a)

A function is continuous at a point a if (1) f(a) is defined, (2) lim(x→a) f(x) exists, and (3) the limit equals f(a).

Worked examples

Is f(x) = (x²-4)/(x-2) continuous at x = 2?
  1. f(2) is undefined (0/0), so condition 1 fails

Answer: No, f is not continuous at x = 2 (not defined there).

Is f(x) = x² + 1 continuous at x = 3?
  1. f(3) = 10 -defined
  2. lim(x→3)(x²+1) = 10 -exists
  3. Limit = f(3) ✓

Answer: Yes, f is continuous at x = 3.

Related formulas

Intermediate Value TheoremExtreme Value Theorem

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