Discover the epsilon-delta definition and properties of continuous functions
Continuity captures the intuitive idea that a function has "no jumps" - small changes in input produce small changes in output. The rigorous epsilon-delta definition states:
For every ε > 0, there exists δ > 0 such that |x - a| < δ implies |f(x) - f(a)| < ε
This section explores what continuity really means and proves powerful theorems like the Intermediate Value Theorem and the Extreme Value Theorem.
Explore the formal definition interactively. Adjust ε to see how δ must respond to keep the function values within the target band.
f is continuous at a if: for every ε > 0, there exists δ > 0 such that
|x - a| < δ ⟹ |f(x) - f(a)| < ε
If the curve stays within the green ε-band whenever x is in the red δ-interval, then δ works for this ε.
Not all discontinuities are created equal. Explore removable, jump, infinite, and essential discontinuities with visual examples.
At x = 1:
Left limit: 2
Right limit: 2
f(1): undefined
The limit exists but f(a) is undefined or ≠ limit. Can be "fixed" by redefining f(a).
This function: Removable discontinuity at x = 1 (equals x + 1 elsewhere)
If a continuous function takes values f(a) and f(b) at the endpoints of an interval, it must take every value between them. Watch the bisection method find roots using this principle.
✓ IVT applies: f(a) and f(b) are on opposite sides of the target!
If f is continuous on [a, b] and y is any value between f(a) and f(b), then there exists some c ∈ (a, b) such that f(c) = y.
The bisection method exploits this: repeatedly halve the interval, keeping the half where the sign change occurs.
Pointwise continuity allows δ to depend on the point x. Uniform continuity requires the same δ to work everywhere. Compare these concepts visually.
The same δ = 0.3030 works for all points in the interval.
Pointwise continuity: For each point x and ε > 0, ∃ δ > 0 (δ may depend on both x and ε)
Uniform continuity: For each ε > 0, ∃ δ > 0 that works for ALL points (δ depends only on ε, not on x)
Key theorem: A continuous function on a closed bounded interval [a, b] is always uniformly continuous.
A continuous function on a closed bounded interval always attains its maximum and minimum values. Explore this fundamental result.
If f is continuous on a closed bounded interval [a, b], then f attains both a maximum and minimum value on that interval.
That is, there exist points c, d ∈ [a, b] such that:
f(c) ≤ f(x) ≤ f(d) for all x ∈ [a, b]
Key insight: The function must be continuous on a closed and bounded interval. Open intervals or discontinuous functions may fail to attain extrema.
If f is continuous on [a, b] and y is between f(a) and f(b), then f(c) = y for some c ∈ (a, b).
If f is continuous on [a, b], then f attains both a maximum and minimum value on that interval.
A continuous function on a closed bounded interval [a, b] is uniformly continuous.
Continuous functions map connected sets to connected sets (no "tearing").