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Additional resources for A note on regularity of solutions to degenerate elliptic equations of Caffarelli-Kohn-Nirenberg type

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The most prevalent example is as x 1 , grows rapx idly toward positive infinity on the right of 0 and toward negative infinity on 1 the left of 0. This rapid growth of causes the sine graph to oscillate wildly, x as shown in Figure 3-4. approaches 0. As this happens, the input to the sine function, 1 y Figure 3-4 x -1 1 -1 43 Continuity on an Interval A function is considered continuous on an interval of its domain if it is continuous at all points in the interval. You can talk about continuity on open, closed, or half-open intervals.

The majority of examples you will encounter in calculus deal with what is happening to a function as the independent variable approaches a number or as the absolute value of the independent variable gets infinitely large. Again, it is critical to understand that it is not necessary for the function take on the limiting value for a limit to exist. It is also important to understand that a limit reports the y value being approached as x changes. The notation looks like lim( 3 x + 1) = 7 . ” In this case, the function 3 x + 1 actually takes on the limit value, seven, at x = 2, but that is not always necessary, as you will see in later examples.

In math, the same common-sense understanding of continuity can help you tell whether a function is continuous or not, but you still need to be able to approach it a bit more formally. You also need to acquire a general understanding of why continuity is important in calculus. Simply stated, a function is continuous at a given point if there is no kind of a break in the function at that point. If this is true, then a single mathematical statement can summarize all the critical elements of this idea.

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A note on regularity of solutions to degenerate elliptic equations of Caffarelli-Kohn-Nirenberg type by Felli V., Schneider S.

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