Limits and Continuity

1. Intuition of Limits

A limit captures the behavior of a function near a point, not necessarily at the point.

lim_{x->a} f(x)=L means values of f(x) can be made arbitrarily close to L by taking x sufficiently close to a.

2. Formal Epsilon-Delta Definition

For every epsilon > 0, there exists delta > 0 such that if 0 < |x-a| < delta, then |f(x)-L| < epsilon.

This is the precise notion used in proofs.

3. One-Sided Limits

  • Left limit: x->a-
  • Right limit: x->a+

Two-sided limit exists iff both one-sided limits exist and are equal.

4. Continuity at a Point

Function f is continuous at a iff: 1. f(a) defined 2. lim_{x->a} f(x) exists 3. lim_{x->a} f(x)=f(a)

5. Types of Discontinuity

  • Removable (hole)
  • Jump
  • Infinite (vertical asymptote)
  • Oscillatory

6. Theorem (Algebra of Limits)

If lim f = L and lim g = M, then: - lim (f+g)=L+M - lim (fg)=LM - lim (f/g)=L/M if M!=0

Proofs follow epsilon-delta bounds.

7. Worked Examples

  1. lim_{x->2} (x^2-4)/(x-2)=4
  2. Piecewise function matching for continuity at boundary
  3. lim_{x->0} sin(x)/x = 1 (fundamental limit)

8. CS Connections

  • continuity assumptions in optimization and gradient methods
  • numerical solvers depend on local behavior
  • non-smooth objectives (ReLU, absolute value) require subgradient handling

Exercises

  1. Prove limit of a linear function using epsilon-delta.
  2. Classify discontinuity type for three piecewise examples.
  3. Construct function with removable discontinuity at x=3.
  4. Explain why continuity matters in bisection root-finding.