Mathematics
Grade 12
15 min
Determine continuity on an interval using graphs
Determine continuity on an interval using graphs
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Introduction & Learning Objectives
Learning Objectives
Identify points of discontinuity (holes, jumps, asymptotes) by visually inspecting a function's graph.
Classify discontinuities as removable, jump, or infinite based on their graphical representation.
Determine if a function is continuous on a given open interval (a, b) by analyzing its graph.
Determine if a function is continuous on a given closed interval [a, b] by analyzing its graph, including the behavior at the endpoints.
Explain why a function is or is not continuous on an interval by referencing the three-part definition of continuity in a graphical context.
Use one-sided limits to describe the behavior of a function at points of discontinuity shown on a graph.
Can you trace the entire path of a roller coaster on a blueprint without ever lift...
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Key Concepts & Vocabulary
TermDefinitionExample
Continuity at a PointA function is continuous at a point if its graph does not have any breaks, holes, or jumps at that point. You can trace the graph through the point without lifting your pencil.On the graph of y = x², the function is continuous at x = 2 because there is a solid point on the curve at (2, 4) and no break in the graph.
Removable DiscontinuityA single point is missing from the graph, represented by an open circle or 'hole'. The limit exists at this point, but the function is either undefined or has a different value.The function f(x) = (x²-4)/(x-2) has a hole at x = 2. The graph looks like the line y = x+2, but with an open circle at (2, 4).
Jump DiscontinuityThe graph 'jumps' from one y-value to another at a specific x-value. The...
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Core Formulas
The Three-Part Definition of Continuity at a Point c
A function f is continuous at a point c if: 1. f(c) is defined. 2. lim_{x→c} f(x) exists. 3. lim_{x→c} f(x) = f(c).
This is the formal test for continuity at a single point. Graphically, condition 1 means there's a solid dot at x=c. Condition 2 means the graph approaches the same y-value from both the left and right. Condition 3 means the solid dot is exactly where the graph is heading.
Continuity on an Open Interval (a, b)
A function f is continuous on (a, b) if it is continuous at every point c where a < c < b.
To check this graphically, scan the portion of the graph strictly between x=a and x=b. If there are no holes, jumps, or vertical asymptotes in that section, the function is continuous on the open interv...
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Challenging
The graph of f(x) has a jump at x=0 and a vertical asymptote at x=3. On which of the following disjoint intervals is the function continuous?
A.[-2, 0) U (0, 3)
B.(-∞, 0] U [3, ∞)
C.(-∞, 0) U (3, ∞)
D.[-2, 0] U (3, 4]
Challenging
The graph of f(x) is a V-shape with its vertex at (2, -1), opening upwards. It is continuous everywhere. Consider the function g(x) = 1/f(x). On which interval would g(x) be discontinuous?
A.g(x) is continuous everywhere.
B.g(x) is discontinuous at x = -1.
C.g(x) is discontinuous at x = 2.
D.g(x) is discontinuous at the x-intercepts of f(x).
Challenging
A function f(x) is graphed on [-5, 5]. It has a jump discontinuity at x = -1 and a removable discontinuity at x = 3. Which of the following is NOT a valid reason for f(x) being discontinuous on the closed interval [-2, 4]?
A.The function is not continuous at x=2, which is in [0, 5].
B.The function is not continuous on the open interval (0, 5).
C.The one-sided limits at x = -3 are not equal.
D.The three-part definition of continuity fails at x=2.
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