Continuity with piece-wise

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...

  1. Left-handed limit: We're using 2x + 5 (since that's the function when x < 0)
  2. Right-handed limit: We're using x2 (since that's the function when x > 0)
  3. At x = 0: We're using x2 (since that's the function when x = 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!

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