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The graph of in Figure 2.1.4(a) shows that this function has a jump discontinuity at .
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Secant lines through for the curve to the left of also pass through . The limiting value of the slopes of these secants is 2.
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Secant lines through for the curve to the right of also pass though , . The limiting value of the slopes of these secants is 1.
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The portion of the graph to the left of has a tangent line with slope 2; but the portion to the right has a tangent line with slope 1.
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plot(piecewise(x<=1,x^2,x*(3-x)),x=-2..4,discont=true,view=[-2..4,-2..3],scaling=constrained);
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Figure 2.1.4(a) Graph of with its jump discontinuity
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The following calculations verify that the limiting slopes of secant lines to either side of differ.
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Limiting slopes
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To the left of
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To the right of
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=
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=
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On the closed interval , the function would be said to have slope 2 at by considering just a one-sided limit at . However, because of the jump at in , it is probably best to say that there is no unique slope for this function at this point.
