Continuity
From the intuitive "drawing without lifting the pen" to the rigorous limit-based definition.
What is Continuity?
A function is continuous at if:
This requires three things: (1) exists, (2) the limit exists, and (3) they are equal.
Taxonomy of Discontinuities
Limit exists, but is undefined or different.
Real World: A circuit with a missing component that can be "bridged."
Left and right limits exist but differ.
Real World: Tax brackets, quantum energy levels, digital logic.
Limit is (vertical asymptote).
Real World: Phase transitions, resonance in circuits.
Pathological Functions: Why Definitions Matter
Why "Drawing with a Pencil" is Insufficient
Some valid functions cannot be drawn at all, forcing us to rely on the rigorous definition.
Discontinuous everywhere. Cannot be drawn, yet it's a valid function.
Continuous at all irrational numbers, discontinuous at all rational numbers. A bizarre yet valid example.
The Intermediate Value Theorem (IVT)
If is continuous on , then for every value between and , there exists some such that .
Application: Root Finding (Bisection Method)
If and , there must be a root where .
This is how calculators solve equations numerically: they narrow the interval until the root is pinned down.
Intermediate Value Theorem Visualizer
For any value N between f(a) and f(b), there exists a point c where f(c) = N.
Key insight: Because f is continuous on [0, 3], the IVT guarantees that for ANY value N between f(0)=1 and f(3)=7, there must exist a point c where f(c) = N. Try moving the slider!
Real World: The Wobbly Table Theorem
A mathematical proof using the IVT shows that a four-legged table on an uneven floor can always be rotated (within 90 degrees) to a stable position where all four legs touch the ground.
The requirement: the ground must be a continuous surface. The IVT guarantees that somewhere in the rotation, the "wobble function" equals zero.
Practice Quiz
Loading...