Limits & Continuity
Defining Continuity
Using the three-part definition of continuity at a point.
Advanced20 min lesson2 min readUpdated August 12, 2026Author not yet attributed
Prerequisites
- Evaluating Limits Algebraically
When a Function's Value Matches What It Approaches
Definition — Continuity at a Point
f is continuous at x = c when three conditions hold: f(c) is defined, limₓ→c f(x) exists, and limₓ→c f(x) = f(c). If any one fails, f is discontinuous at c.
Worked Example — Identifying a Removable Discontinuity
Let h(x) = (x² − 25)/(x − 5) for x ≠ 5, with h(5) defined separately as 12. Since h(x) = x + 5 for x ≠ 5, limₓ→5 h(x) = 10 — the limit exists. But h(5) = 12 ≠ 10, so h is discontinuous at x = 5: a removable discontinuity, since redefining h(5) = 10 would patch it.
Definition — Intermediate Value Theorem
If f is continuous on [a, b] and k is any value between f(a) and f(b), then some c in (a, b) satisfies f(c) = k.
Worked Example — Applying the Intermediate Value Theorem
Show that f(x) = x³ − 2x − 5 has a zero in (2, 3). f is a polynomial, continuous everywhere. f(2) = 8 − 4 − 5 = −1, and f(3) = 27 − 6 − 5 = 16. Since f(2) < 0 < f(3) and f is continuous on [2, 3], the Intermediate Value Theorem guarantees a zero in (2, 3), without solving the cubic exactly.
Tip
Check whether the function is continuous on the whole interval before invoking IVT — a discontinuous function can skip over a target value entirely.
Common Mistakes
Treating 'f(c) is defined' as sufficient for continuity at c.
All three conditions are required together — f(c) defined, the limit existing, and the two matching. h(x) above shows a function that's defined at a point but still discontinuous there.
Key Takeaways
- Continuity requires f(c) defined, limₓ→c f(x) to exist, and the two to be equal.
- A removable discontinuity is where the limit exists but disagrees with (or the function lacks) a matching value at that point.
- The Intermediate Value Theorem guarantees a continuous function hits every value between two endpoint values.
Summary
This closes the study of limits and continuity. The next unit turns that approaching-behavior into a new object — the derivative.
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