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Daily Math Minute

Unit 1: Limits & Continuity

Continuity

Determining continuity at a point and applying the Intermediate Value Theorem.

Advanced25 min lesson2 min readUpdated August 12, 2026Author not yet attributed

Prerequisites

  • Evaluating Limits

When a Function's Value Matches What It Approaches

Definition — Continuity

f is continuous at x = c when f(c) is defined, limₓ→c f(x) exists, and the two are equal. A discontinuity is removable (the limit exists but disagrees with, or f(c) is undefined), a jump (finite one-sided limits that disagree), or infinite (a one-sided limit is unbounded).

Worked Example — Classifying a Removable Discontinuity

Let h(x) = (x² − 16)/(x − 4) for x ≠ 4, with h(4) defined separately as 3. Since h(x) = x + 4 for x ≠ 4, limₓ→4 h(x) = 8. But h(4) = 3 ≠ 8, so h is discontinuous at x = 4 — removable, since redefining h(4) = 8 would patch it.

A jump discontinuity looks different: both one-sided limits exist and are finite, but disagree with each other (the graph steps to a new level). An infinite discontinuity has at least one one-sided limit diverging to ±∞ (a vertical asymptote). Continuity guarantees something powerful about a function's graph over an interval.

Definition — Intermediate Value Theorem

If f is continuous on [a, b] and k is any value between f(a) and f(b), some c in (a, b) satisfies f(c) = k.

Worked Example — Applying the Intermediate Value Theorem

Show f(x) = x³ − 4x − 1 has a zero in (2, 3). f is a polynomial, continuous everywhere. f(2) = 8 − 8 − 1 = −1, f(3) = 27 − 12 − 1 = 14. Since f(2) < 0 < f(3) and f is continuous on [2, 3], IVT guarantees a zero in (2, 3), without solving the cubic.

Tip

IVT requires continuity on the whole interval first — check that (or know it, as with any polynomial) before invoking the guarantee.

Common Mistakes

  • Treating 'f(c) is defined' as sufficient for continuity at c.

    All three conditions are required together — a function can be defined at c and still fail to be continuous there, as h(x) above shows.

Key Takeaways

  • Continuity requires f(c) defined, the limit existing, and the two matching.
  • Removable, jump, and infinite discontinuities are distinguished by what the one-sided limits do.
  • IVT guarantees a continuous function hits every value between two endpoint values.

Summary

This closes Unit 1. Unit 2 turns this approaching-behavior into a new object — the derivative — starting from its formal limit definition.

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