The Bridge Analogy
Imagine you are driving a car along the function graph. For the road to be safe (continuous), three things must be true:
- There must be a bridge (The function is defined).
- The roads on both sides must line up (The limit exists).
- The bridge must be at the same height as the road (The value equals the limit).
If any of these fail, you have a discontinuity. Use the lab below to repair the broken bridge by aligning the function value with the limit.
Bridge Repair Game
DISCONTINUOUS
Continuity Checklist
1. Defined
f(2) exists.
f(2) exists.
2. Limit Exists
Left/Right meet at y=3.
Left/Right meet at y=3.
3. They Match
0.0 != 3.0
0.0 != 3.0
Formal Definition
A function is continuous at a point if and only if:
Types of Discontinuities
[Image of types of discontinuity graphs]Removable (Hole)
The limit exists, but there's a hole in the graph. This is what you fixed in the lab above.
Jump
The left and right limits don't match. The graph snaps to a new elevation.
Infinite
The function explodes to infinity (Vertical Asymptote). The road hits a wall.