Mathematics
Grade 12
15 min
Determine continuity using graphs
Determine continuity using graphs
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1
Introduction & Learning Objectives
Learning Objectives
Define continuity at a point by observing its graphical representation.
Identify the x-values where a function is discontinuous by inspecting its graph.
Classify discontinuities seen on a graph as removable, jump, or infinite.
Relate the three graphical conditions for continuity to the formal limit definition of continuity.
Determine the intervals over which a function is continuous using its graph.
Explain why a function is discontinuous at a specific point by referencing its graphical features like holes, jumps, or asymptotes.
Can you trace the path of a roller coaster on a piece of paper without ever lifting your pencil? 🎢 That simple idea is the visual key to understanding continuity in calculus!
This tutorial will teach you how to determine if a fu...
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Key Concepts & Vocabulary
TermDefinitionExample
Continuity at a PointA function is continuous at a point if its graph is unbroken at that point. Intuitively, you can draw the graph through that point without lifting your pencil.The graph of the parabola f(x) = x^2 is continuous at x = 2 because there is no hole, jump, or gap at that point.
DiscontinuityA point on the graph where the function is not continuous. This appears as a break in the graph.The function f(x) = 1/x has a discontinuity at x = 0 because the graph breaks and goes to infinity on either side.
Removable Discontinuity (Hole)A type of discontinuity where the graph is continuous everywhere except for a single missing point, which is represented by an open circle.The graph of f(x) = (x^2 - 9)/(x - 3) looks like the line y = x + 3, but with a hole at x...
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Core Formulas
The Graphical Conditions for Continuity
A function f is continuous at x = c if and only if: 1. f(c) is defined (there is a solid dot). 2. lim_{x->c} f(x) exists (the graph approaches the same y-value from the left and right). 3. lim_{x->c} f(x) = f(c) (the point approached is the same as the solid dot).
This is the formal definition translated into three visual checks. To confirm continuity at a point on a graph, you must verify that all three of these conditions are met.
Classifying Discontinuities Graphically
If lim_{x->c^-} f(x) ≠lim_{x->c^+} f(x), it's a Jump. If lim_{x->c} f(x) exists but ≠f(c) (or f(c) is undefined), it's Removable. If lim_{x->c} f(x) = ±∞, it's Infinite.
Use these limit-based observations to classify the type of brea...
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Challenging
The graph of a piecewise function is shown. One piece ends in an open circle at (2, k). The second piece begins with a solid dot at (2, 4) and continues to the right. For the entire function to be continuous at x = 2, what must be the y-coordinate of the open circle?
A.k = 2
B.k = 0
C.k can be any value
D.k = 4
Challenging
A function's graph is known to be continuous on the closed interval [0, 5]. Which of the following graphical features is IMPOSSIBLE to find on the graph within the interval (0, 5)?
A.sharp corner at x = 2.
B.vertical tangent at x = 3.
C.hole at x = 4.
D.local maximum at x = 1.
Challenging
The graph of f(x) has a jump discontinuity at x = -1, where lim_{x->-1^-} f(x) = 3 and lim_{x->-1^+} f(x) = -2. The point (-1, 3) is a solid dot on the graph. Which statement correctly describes the continuity of f(x) at x = -1?
A.It is continuous from the left.
B.It is continuous from the right.
C.It is continuous at x = -1.
D.It is neither continuous from the left nor from the right.
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