Module 3 · Lesson 3.4

Continuity — Where Curves Don't Break

Recognize continuity as a curve you can draw without lifting the pen, and check the three conditions that guarantee a function is continuous at a point.

Intuition

Imagine tracing a graph with a pen. If you never have to lift the tip, the curve is continuous. A jump, a hole, or a vertical asymptote forces you to lift the pen; those are the classic discontinuities.

The same idea appears in network signals: a continuous stream of packets arrives without dropout. A sudden gap or an undefined reading is a discontinuity in the transmission. Continuity is the mathematical guarantee that nothing “breaks” at the point you care about.

Core concept

A function is continuous at $x=a$ when three things hold at once: the function is defined there, the limit exists there, and the limit equals the function value. Fail any one of those and the pen has to lift.

Point$x=2$
$f(2)$ defined?Yes, $f(2)=3$
$\lim_{x\to 2}f(x)$Exists, $=3$
Limit $=f(2)$?Yes
VerdictContinuous at $2$

Interactive Lab

Choose a function type and watch the three continuity checks update live. The graph shows whether the curve is unbroken at the marked point.

Continuous
x y
$f(a)$ is defined
$\lim_{x\to a}f(x)$ exists
$\lim_{x\to a}f(x)=f(a)$

All three conditions must hold for continuity at the marked $x$-value. A single failure produces a break—exactly the kind of dropout a network monitor would flag.

Key Idea

$$ \lim_{x \to a} f(x) = f(a) $$

This single equation packs the three requirements: the limit on the left must exist (so left- and right-hand limits agree), $f(a)$ on the right must be defined, and the two sides must be equal. When the equation holds, the graph has no break at $x=a$ and can be drawn without lifting the pen.

$a$The $x$-value where we test continuity
$f(a)$The actual function value at that point (must exist)
$\lim_{x\to a}f(x)$The height the curve approaches from both sides
$=$Equality of limit and value guarantees no break

Think Further

  1. Why does a removable hole fail the continuity test even though the limit exists?
  2. In a network signal, which of the three continuity conditions corresponds to a sudden packet dropout that never resumes?
  3. Can a function be continuous at every point in an open interval yet still have a discontinuity at an endpoint? Sketch the idea.
Show suggested answers
  1. Continuity also requires the function value to exist and equal the limit. At a removable hole, that value is missing or incorrect.
  2. The function value or the relevant limit fails to exist, depending on how the dropout is modeled.
  3. Yes. The function may be continuous at every interior point but undefined, jump, or approach the wrong value at an endpoint.