2.5 Continuity and the Intermediate Value Theorem - Quick Reference

Continuity of a Function

• Definition: A function f is continuous at a number a if $\lim_{x\to a} f(x) = f(a).$
• Three requirements implicit in the definition:
  • f(a) is defined
  • $\lim_{x\to a} f(x)$ exists
  • $\lim_{x\to a} f(x) = f(a)$
• Intuition: small changes in x produce small changes in f(x).
• Graphical meaning: a continuous function has no breaks; its graph can be drawn without lifting the pen.
• Continuity on an interval: f is continuous at every number in the interval; endpoints use one-sided continuity

Right/Left Continuity and Endpoints

• Right-continuous at a: $\lim_{x\to a^+} f(x) = f(a).$
• Left-continuous at a: $\lim_{x\to a^-} f(x) = f(a).$
• Continuous on an interval means continuous at every point in the interval; at endpoints, continuity is understood as one-sided.

Discontinuities

• f is discontinuous at a if it fails the continuity condition at a.
• Criteria for discontinuity: either f(a) not defined, or the limit does not exist, or the limit exists but differs from f(a).
• Graphically, discontinuities appear as breaks, holes, or jumps in the graph.

Types of Discontinuities

• Removable: a hole at x = a; can be fixed by redefining f(a) to the limit as x -> a.
• Infinite: limit diverges to ±∞ (vertical asymptote).
• Jump: left and right limits exist but are not equal.
• Other example: greatest integer function has discontinuities at all integers.

Continuous Functions – Quick Facts

• A function is continuous on an interval if it is continuous at every point of the interval.
• Endpoints: continuity at an endpoint refers to the appropriate one-sided limit (right at the left endpoint, left at the right endpoint).

Theorems: Building Continuous Functions

• Theorem (Closed under basic operations): If f and g are continuous at a and c is a constant, then the following are continuous at a:
  • f + g
  • f − g
  • c f
  • f g
  • if g ≠ 0 near a, then f/g is continuous at a

Common Continuous Functions

• If f and g are continuous on an interval, then so are f + g, f − g, cf, fg, and (under the condition on g) fg/(g) (i.e., quotients where defined).
• In particular:
  • Polynomials are continuous everywhere: $f(x) = a<em>n x^n + \cdots + a</em>1 x + a_0.$
  • Rational functions are continuous wherever they are defined (i.e., denominator ≠ 0).

Examples of Continuity in Practice

• Volume V of a sphere: $V(r) = \frac{4}{3}\pi r^3$ is a polynomial in r, hence continuous for all r.
• Height of a vertically thrown ball: $h(t) = -16t^2 + v<em>0 t + h</em>0$ is a polynomial in t, hence continuous in t.
• Trigonometric and exponential functions are continuous on their domains; compositions of continuous functions are continuous.

Composite Functions

• If f is continuous at b and lim{x\to a} g(x) = b, then $\lim</em>{x\to a} f(g(x)) = f(b).$
• If g is continuous at a and f is continuous at g(a), then the composite function $f\circ g$ is continuous at a.

The Intermediate Value Theorem (IVT)

• If f is continuous on [a, b] and f(a) ≠ f(b), then for any N between f(a) and f(b) there exists c in (a, b) with $f(c) = N.$ (Not true for discontinuous functions.)
• Intuition: a continuous graph on [a,b] cannot jump over a horizontal level y = N.
• Applications: use IVT to prove existence of roots and to bracket values; it underpins root-finding methods.

Examples and Applications of IVT

• If a polynomial f satisfies f(1) < 0 and f(2) > 0, then there exists c in (1, 2) with f(c) = 0.
• By refining interval (e.g., 1.2 to 1.3), one can locate a root to any desired accuracy.
• Computational graphs rely on IVT principles: assuming continuity, connecting plotted points approximates the true curve.

Special Notes

• Sine and cosine are continuous everywhere (with sin 0 = 0, cos 0 = 1).
• Tan x is continuous except where cos x = 0, i.e., at x = (π/2) + kπ, which are infinite discontinuities.
• One-sided continuity and continuity on an interval provide a foundation for more advanced results in calculus.