Continuity with piece-wise
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A piece-wise function has different functions across different intervals of x-values.
Scenario: Verify if the following function is continuous.
It's at x = 0 where our function switches... so this is where we'll test for continuity!
Notice that...
- Left-handed limit: We're using
2x + 5(since that's the function whenx < 0) - Right-handed limit: We're using
x2(since that's the function whenx > 0) - At x = 0: We're using
x2(since that's the function whenx = 0)
Answer: The function is not continuous at x = 0, since the left-handed limit of y = 5 does not equal the right-handed limit or value at the x-value of y = 0.
In other words... at x = 0 the function has a drop-off from y = 0 to y = 5!