Lesson 3: Continuity & Intermediate Value Theorem

1. What is Continuity?

A function is continuous at a point if:

1. \(f(a)\) is defined
2. \(\lim_{x→a} f(x)\) exists
3. \(\lim_{x→a} f(x) = f(a)\)

Intuition: no jumps, breaks, or holes in the graph at that point.

Polynomials, \(\sin(x)\), \(e^{x}\), rational functions (where denominator ≠ 0) are continuous.

2. Intermediate Value Theorem (IVT)

If f is continuous on \([a,b]\) and \(k\) is between \(f(a)\) and \(f(b)\), then there exists \(c\) in \((a,b)\) such that \(f(c) = k\).

Intuition: continuous function takes every value between its min and max on an interval.

Used to prove existence of roots.

Summary – Lesson 3

• Continuous = no jumps/holes
• Polynomials and most common functions are continuous
• IVT: continuous function takes every value between \(f(a)\) and \(f(b)\)

Next: the derivative – slope of tangent line.