AP Calculus AB/BC

Dec 7, 2025

2 pages

Understanding Continuity and One-Sided Limits

Chelsea @zuncho

Calculus continuity might sound complex, but it's really about understanding when functions behave smoothly without breaks or jumps.... Show more

# Calculus Section 1.4 Continuity and One-Sided Limits
- Determine continuity at a point and continuity on an open interval.
- Determine one

Continuity at a Point and on an Open Interval

Ever wonder if you could draw a function without lifting your pencil? That's essentially what continuity is all about! A continuous function has no holes, jumps, or gaps in its graph.

For a function to be continuous at a point c, three conditions must be met f(c) must be defined, the limit of f(x) as x approaches c must exist, and that limit must equal f(c). If a function meets these conditions at every point within an interval (a,b), we say it's continuous on that interval.

Not all functions are continuous everywhere. For example, f(x) = 1/x isn't continuous at x = 0 because division by zero isn't defined. Piecewise functions might have discontinuities where the pieces meet, and rational functions can have issues where the denominator equals zero.

Remember This Discontinuities come in two flavors - removable (where we could "fix" the function by redefining it at a single point) and nonremovable (where no single value can make the function continuous at that point).

When checking a function for continuity, always work through all three conditions systematically. Many functions you'll encounter in basic calculus $like polynomials and sine/cosine$ are continuous everywhere, which is why they're so useful for modeling real-world situations.

One-Sided Limits

Sometimes we need to approach a point from just one direction - that's where one-sided limits come in handy. The right-hand limit approaches from values greater than the point, while the left-hand limit approaches from values less than the point.

One-sided limits are particularly useful for piecewise functions and graphs with jumps. For a regular $two-sided$ limit to exist, both the left-hand and right-hand limits must exist and be equal. If they're different, the overall limit doesn't exist, but we can still describe what happens from each side.

For a function to be continuous on a closed interval $a,b$ , it needs to be continuous on the open interval (a,b), continuous from the right at a, and continuous from the left at b. This extension allows us to work with functions on bounded domains.

Pro Tip When evaluating limits of rational functions that give 0/0, try factoring the numerator and denominator to cancel common terms. This technique often reveals the actual limit value, as seen in the example where lim(x→3) $x-3$ / $3x²-9$ = 1/6.

When tackling limit problems, drawing a quick sketch of the function can help you visualize the behavior from both sides of the point in question. This visual approach often makes the concept of one-sided limits much clearer.

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AP Calculus AB/BC

•

Dec 7, 2025

•

2 pages

Understanding Continuity and One-Sided Limits

Chelsea

@zuncho

Calculus continuity might sound complex, but it's really about understanding when functions behave smoothly without breaks or jumps. This concept is crucial because it helps determine where mathematical models accurately represent real-world situations and where they might fail.

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Continuity at a Point and on an Open Interval

Ever wonder if you could draw a function without lifting your pencil? That's essentially what continuity is all about! A continuous function has no holes, jumps, or gaps in its graph.

For a function to be continuous at a point c, three conditions must be met: f(c) must be defined, the limit of f(x) as x approaches c must exist, and that limit must equal f(c). If a function meets these conditions at every point within an interval (a,b), we say it's continuous on that interval.

Remember This: Discontinuities come in two flavors - removable (where we could "fix" the function by redefining it at a single point) and nonremovable (where no single value can make the function continuous at that point).

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One-Sided Limits

Pro Tip: When evaluating limits of rational functions that give 0/0, try factoring the numerator and denominator to cancel common terms. This technique often reveals the actual limit value, as seen in the example where lim(x→3) $x-3$ / $3x²-9$ = 1/6.