Skill goal: Decide whether a function is continuous at a point using the three-part test.
1 · Observe
Meet the function h(x):
h(x) = x for x < 1, h(x) = x + 2 for x ≥ 1
The graph on [−1, 3] is below. Notice the open circle at (1, 1) and the filled dot at (1, 3).
2 · Manipulate
Drag the blue probe along the curve and watch x and h(x).
Or let the probe approach x = 1 automatically:
3 · Predict
Before looking at Section 4, type your guesses, then press Check my prediction.
4 · See
Live readouts for the probe point x = 1:
Coming from the left, values slide toward 1. Coming from the right, they slide toward 3. The two one-sided limits disagree — so the two-sided limit does not exist.
5 · Explain
A function f is continuous at c exactly when all three conditions hold:
- f(c) is defined. Here h(1) = 3 ✓ (filled dot).
- The limit exists: left-hand limit = right-hand limit. Here 1 ≠ 3 ✗ — this condition fails.
- limit = f(c). (Never reached, since the limit doesn’t exist.)
6 · Challenge
Three rounds. Decide: continuous at the point? If not, name the discontinuity type.
Score: 0 / 3