Continuity quiz #1 Flashcards
Continuity quiz #1
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What is the formal definition of continuity for a function f(x) at a point c?
A function f(x) is continuous at a point c if the limit as x approaches c of f(x) equals the function value at c; that is, limₓ→c f(x) = f(c). How can you visually determine if a function is continuous at a specific point using its graph?
You can visually determine continuity at a point by tracing the graph with a pen; if you can do so without lifting your pen at that point, the function is continuous there. Where are rational functions typically discontinuous, and how can you find these points?
Rational functions are typically discontinuous where their denominator equals zero. To find these points, set the denominator equal to zero and solve for x. How do you check for continuity at the boundary point of a piecewise function?
To check continuity at the boundary of a piecewise function, compare the left-hand and right-hand limits at the boundary point and the function value there. If all are equal, the function is continuous at that point; otherwise, it is discontinuous. What is the formal definition of continuity for a function f(x) at a point c?
A function f(x) is continuous at c if limₓ→c f(x) = f(c); that is, the limit as x approaches c equals the function value at c. How can you visually determine if a function is continuous at a specific point using its graph?
If you can trace the graph through the point without lifting your pen, the function is continuous at that point. Where are rational functions typically discontinuous, and how can you find these points?
Rational functions are discontinuous where their denominator equals zero; set the denominator to zero and solve for x to find these points. How do you check for continuity at the boundary point of a piecewise function?
Compare the left-hand and right-hand limits at the boundary and the function value; if all are equal, the function is continuous there. What types of discontinuities can appear on a graph, and how do they look?
Discontinuities can appear as holes, jumps, or asymptotes; holes are missing points, jumps are sudden changes in value, and asymptotes are unbounded behavior. What steps do you take to determine if a function is continuous at a specific value c?
Find the limit of the function as x approaches c and the function value at c; if they are equal, the function is continuous at c.
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