Chapter 1: Limits
Section 1.6: Continuity
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Example 1.6.6
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If and , show that the rule for is continuous at even though neither nor is continuous at .
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Solution
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In Figure 1.6.6(a), is graphed in blue, is graphed in green, and their sum, , is graphed in red. From this Figure, it should be clear that both and have jump discontinuities at , but that the rule for the sum does not.
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To be very clear: The rule for the sum does not have a discontinuity at , but the sum of and is defined as a function only on the domain common to both and . This common domain does not include , so the function that is their sum cannot have in its domain. But looking at just the rule for the sum, independent of whence it came, that rule can be assigned the largest possible domain for which it is defined, and that largest domain is the set of all the real numbers.
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>
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unassign('f','g'):
F:=piecewise(x<1,x-1,5):
G:=piecewise(x<1,2*x,2-5*x):
plot([F,G,F+G], x=-2..3, discont=true, color=[blue,green,red],legend = [typeset(f(x)),typeset(g(x)),typeset(f(x)+g(x))]);
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Figure 1.6.6(a) Graphs of , and
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Show is discontinuous at
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Show is discontinuous at
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= but =
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= but =
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Show is continuous at
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Evaluate at
Context Panel: Evaluate and Display Inline
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=
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Expression palette: Limit operator
Context Panel: Evaluate and Display Inline
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=
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