Topic 1.11 key terms
- continuous at a point
- f is continuous at x = c when f(c) exists, the limit of f(x) as x approaches c exists, and that limit equals f(c).
- one-sided limit
- The height f(x) heads toward as x approaches c from one side only: from the left (x → c⁻) or from the right (x → c⁺).
- jump discontinuity
- Both one-sided limits exist but are different, so the limit does not exist. No single value at c can make f continuous there.
- removable discontinuity
- The limit at c exists, but f(c) is missing or different from it. Defining or redefining f(c) as the limit removes the discontinuity.
- infinite discontinuity
- A discontinuity due to a vertical asymptote: at least one one-sided limit is infinite, so no value of f(c) can make f continuous.
- removing a discontinuity
- If the limit exists at a discontinuity, define or redefine the function's value there to equal the limit.
- piecewise boundary condition
- At a boundary, a piecewise function is continuous when the expressions on both sides give the same value there and that value is the function's value at the boundary.
Transcript
374 words · 1 parts
Continuous, exactly
We’ve now seen three ways a function can break at one point, so let’s line them up and look at exactly what went wrong in each one. In the first one, the formula has no value at zero at all. In the jets’ longitude, the two sides head to different heights, so there’s no limit. With the lazy patch, there’s a value and there’s a limit, but they don’t match.
Each of those failures hands us one piece of the definition. A function f is continuous at x equals c when three things are all true. First, f of c exists, so there’s an actual value there.
Second, the limit of f as x approaches c exists, which means both sides head to the same height. Third, that limit equals f of c, so the value sits exactly where the nearby values are heading.
You can pack all three into one line, the limit of f as x approaches c equals f of c. Whatever the nearby values promise, the function actually delivers. When an exam question says justify that a function is continuous, this is what it wants to see, all three pieces, written out.
There’s one more kind of break you need to know. Look at one over x minus two, squared, near x equals two. At one point nine it’s a hundred, at one point nine nine it’s ten thousand, and from the right side you get the very same numbers. The closer you get to two, the bigger it gets, without any bound at all, so there’s no height for the two sides to head toward.
That’s an infinite discontinuity, the kind with a vertical asymptote, and no number you write at x equals two can ever fix it, because no number is big enough. Let’s put the three kinds side by side and sort them by one question, can a single value fix it? You can repair the removable kind with one value, because both sides already approach the same finite height. Put the value at that height and the graph joins, just like the laser’s seam.
Try this one. This graph has four marked inputs, and I want you to find the one where f is actually continuous.