An unbroken line from −∞ to +∞

continu.ax

f : ℝ → ℝ

1

i.

Continuity

A function is continuous at a point when the limit exists, equals the value, and the line does not break. No gaps. No jumps. Only smooth, unbroken progression.

lim_x→a f(x) = f(a)

2

ii.

The Interval

Between any two points on the real line lies an infinite set. Continuity means traversing every one of them. No skipping. No teleportation. Only presence.

∀ ε > 0, ∃ δ > 0

3

iii.

Rate of Change

The derivative measures how a continuous function changes. At every point along the axis there is a slope, a direction, a tendency. Continuity is what makes calculus possible.

f′(x) = lim_h→0 [f(x+h) − f(x)] / h

4

iv.

The Integral

Integration sums the continuous. Every infinitesimal slice contributes to the whole. The area under the curve is the long, quiet story of accumulation.

∫_a^b f(x) dx

5

v.

Toward Infinity

The axis extends forever. There is no end, only continuation. Each step forward reveals another step, another point, another possibility on the line.

x → ∞

∞

vi.

The Limit

Approaching but never arriving. The asymptote is the horizon of mathematics — always visible, never reached. The journey itself is the destination.

lim_x→∞ f(x) = L