Mathematics
Grade 12
15 min
Find and analyze points of discontinuity using graphs
Find and analyze points of discontinuity using graphs
What you'll learn
- Identify at least 3 different colors (red, blue, yellow) when shown an object.
- Sort a group of 5 or more objects into color groups with 80% accuracy (e.g., placing all red blocks in one pile, all blue blocks in another).
- Point to all the objects of a specific color (e.g., "Point to all the blue things") in a group of 10 mixed-color objects.
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1
Introduction & Learning Objectives
Learning Objectives
Visually identify removable, jump, and infinite discontinuities on a graph.
Determine the x-values where a function is discontinuous by inspecting its graph.
Classify the type of discontinuity at a specific point on a graph.
Analyze and state the one-sided limits (from the left and right) at a point of discontinuity.
Justify why a function is discontinuous at a point by referencing the three-part definition of continuity.
Describe the behavior of a function near an infinite discontinuity using limit notation.
Ever tried to trace a complex drawing without lifting your pencil? ✏️ The points where you're forced to lift it are the focus of our lesson today!
This tutorial explores the concept of continuity from a visual perspective. We will learn how to...
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Key Concepts & Vocabulary
TermDefinitionExample
Continuity at a PointA function f(x) is continuous at a point x=c if its graph is unbroken at that point. Formally, this means the limit as x approaches c exists, the function is defined at c, and the limit equals the function's value.The graph of y = x² is continuous at x=3 because you can trace it through that point without lifting your pencil. The limit as x approaches 3 is 9, and the function's value f(3) is also 9.
Point of DiscontinuityAn x-value where the graph of a function has a break, hole, or gap. At this point, the function fails at least one of the conditions for continuity.The function f(x) = 1/x has a point of discontinuity at x=0, where there is a vertical asymptote.
Removable DiscontinuityA type of discontinuity that appears as a single �...
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Core Formulas
The Three-Part Definition of Continuity
A function f(x) is continuous at x=c if:
1. f(c) is defined.
2. lim_{x→c} f(x) exists.
3. lim_{x→c} f(x) = f(c).
Use this as a formal checklist to prove or disprove continuity at a point. If any one of these three conditions fails, the function is discontinuous at x=c. This is the foundation for all analysis.
Classifying Discontinuities with Limits
Given a discontinuity at x=c:
- Removable if lim_{x→c⁻} f(x) = lim_{x→c⁺} f(x) but this value ≠ f(c) (or f(c) is undefined).
- Jump if lim_{x→c⁻} f(x) ≠ lim_{x→c⁺} f(x) (and both are finite).
- Infinite if lim_{x→c⁻} f(x) = ±∞ or lim_{x→c⁺} f(x) = ±∞.
After visually identifying a discontinuity, use the behavior of the one-sided limits to formally classify its type. This connects the vi...
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Challenging
The graph of a function f(x) is a parabola y = x² with a hole at the point (2, 4). How could the function be defined or redefined at x=2 to make it continuous at that point?
A.Define f(2) = 0
B.Define f(2) = 2
C.Define f(2) = 4
D.It is impossible to make it continuous.
Challenging
A function f(x) is defined and graphed on the closed interval [-5, 5]. At the left endpoint, x=-5, the graph shows a solid dot at (-5, 2) and the curve extends to the right from this point. Which statement is true about the continuity at x=-5?
A.It is discontinuous because the left-sided limit does not exist.
B.It is continuous because lim_{x→-5⁺} f(x) = f(-5).
C.It is discontinuous because the two-sided limit does not exist.
D.It is impossible to determine continuity at an endpoint.
Challenging
The graph of a function f(x) has a vertical asymptote at x=3, where both sides of the graph approach +∞. Which of the following statements about the limit at x=3 is formally incorrect?
A.The behavior can be described as lim_{x→3} f(x) = ∞.
B.The limit as x approaches 3 exists.
C.The function has an infinite discontinuity at x=3.
D.The limit as x approaches 3 from the left does not exist as a finite number.
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Find and analyze points of discontinuity using graphs is a Grade 12 Mathematics lesson on ExcelOS.
What will I learn in Find and analyze points of discontinuity using graphs?
You'll be able to: Identify at least 3 different colors (red, blue, yellow) when shown an object; Sort a group of 5 or more objects into color groups with 80% accuracy (e.g., placing all red blocks in one pile, all blue blocks in another); Point….
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This lesson includes 25 practice questions across multiple difficulty levels, each with instant feedback and explanations.