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Unit 1: Limits & Continuity

Defining Continuity

Using the definition of continuity and classifying discontinuities.

Advanced25 min lesson4 min readUpdated August 12, 2026Author not yet attributed

Prerequisites

  • Limit Laws & Algebraic Techniques

When a Limit and a Function's Value Actually Agree

The very first example of this unit had a limit (2) at a point (x = 1) where the function itself wasn't even defined. Before reading on: what three separate things would all have to be true for a graph to have no break, no jump, and no hole at a point?

Definition — Continuity at a Point

f is continuous at x = c when three conditions all hold: f(c) is defined, limₓ→c f(x) exists, and limₓ→c f(x) = f(c). If any one of these fails, f is discontinuous at c.
lim⁡x→cf(x)=f(c)\lim_{x \to c} f(x) = f(c)

Worked Example — Identifying a Removable Discontinuity

Let h(x) = (x² − 9)/(x − 3) for x ≠ 3, with h(3) defined separately as 10. Since h(x) = x + 3 for x ≠ 3, limₓ→3 h(x) = 3 + 3 = 6 — the limit exists. But h(3) = 10, and 6 ≠ 10, so the third continuity condition fails: h is discontinuous at x = 3, even though both h(3) and the limit individually exist. Because the limit exists and simply doesn't match the function's value, this is a removable discontinuity — redefining h(3) = 6 would patch the hole and make h continuous there.
FunctionBehavior at the pointType of discontinuity
h(x) above, at x = 3Limit exists (6), but h(3) = 10 ≠ 6Removable — a single-point hole
p(x) = 2x (x ≤ 1), x + 4 (x > 1), at x = 1Left-hand limit 2, right-hand limit 5 — finite but unequalJump — the graph steps to a new level
q(x) = 1/(x − 2), at x = 2As x → 2⁻, q(x) → −∞; as x → 2⁺, q(x) → +∞Infinite — a vertical asymptote
The three kinds of discontinuity, distinguished by what the one-sided limits do.

Function Explorer

Transform: g(x) = a·f(b(x − h)) + k
1
1
0
0
Composition
Analysis (of the transformed function, in view)
y-intercept
(0, 0)
x-intercepts
(-9.42, 0), (-6.28, 0), (-3.14, 0), (0, 0), (3.14, 0), (6.28, 0), (9.42, 0)
Extrema
local min at (-7.85, -1); local max at (-4.71, 1); local min at (-1.57, -1); local max at (1.57, 1); local min at (4.71, -1); local max at (7.85, 1)
Inflection points
(-9.42, 0), (-6.28, 0), (-3.14, 0), (0, 0), (3.14, 0), (6.28, 0), (9.42, 0)
Vertical asymptotes
none found in view
Horizontal asymptotes
none found
Domain
all real numbers in view
Range (estimated)
approximately [-1, 1]

Continuity guarantees something powerful about a function's graph: if it's continuous on a closed interval, the graph can't skip over any value between its endpoint values without actually passing through it.

Definition — Intermediate Value Theorem

If f is continuous on [a, b] and k is any value between f(a) and f(b), then there exists at least one c in (a, b) with f(c) = k. Continuity is essential here — a jump discontinuity could skip over k entirely.

Worked Example — Applying the Intermediate Value Theorem

Show that f(x) = x³ − x − 1 has a zero somewhere in (1, 2). f is a polynomial, so it's continuous everywhere, including [1, 2]. Evaluate the endpoints: f(1) = 1 − 1 − 1 = −1, and f(2) = 8 − 2 − 1 = 5. Since f(1) < 0 < f(2) and f is continuous on [1, 2], the Intermediate Value Theorem guarantees some c in (1, 2) with f(c) = 0 — a zero exists there, even without solving the cubic exactly.

Tip

To classify a discontinuity, check what the one-sided limits actually do: both finite and equal but disagreeing with f(c) (or f(c) undefined) is removable; both finite but unequal is a jump; either one going to ±∞ is infinite.

Common Mistakes

  • Concluding the Intermediate Value Theorem applies without first confirming the function is continuous on the interval.

    The theorem's guarantee depends entirely on continuity — a discontinuous function can skip over a target value without ever equaling it, so continuity must be checked (or already known, as with polynomials) before invoking IVT.

  • Treating 'f(c) is defined' as enough by itself to conclude f is continuous at c.

    All three conditions are required together — f(c) defined, the limit existing, and the two agreeing. A function can be defined everywhere and still fail to be continuous at a point, as h(x) above shows.

Key Takeaways

  • Continuity at a point requires f(c) to be defined, limₓ→c f(x) to exist, and the two to be equal.
  • Discontinuities fall into three types — removable, jump, and infinite — distinguished by what the one-sided limits do.
  • The Intermediate Value Theorem guarantees a continuous function hits every value between two endpoint values, useful for proving a zero (or any target value) exists without solving for it exactly.

Summary

This closes Unit 1: limits and continuity describe a function's local behavior precisely, using only the language of 'approaching.' Unit 2 turns that same approaching-behavior into a new object entirely — the derivative — starting from its formal definition as a limit.

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