When we say the limit exists, we mean that the function has a limit around that point and is equal.
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chapter9 Continuity
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Continuity of function: When we draw a function image, the pen tip does not need to leave the paper, and the stroke is completed in one go.
The two functions in Figure 9.1 are continuous, and the two functions in Figure 9.2 have a breakpoint at x=2, so they are not continuous.
3 Conditions of Continuity
for example:
Example 1:
The function has a domain and a limit at x = 2, but the value of the limit is not equal to the value of the function, so the condition (3) is not satisfied, so at x = 2, there is no continuity.
Example 2:
The function has no limit at x=2, because the left and right limits at this point are not equal, so the function is discontinuous at x=2.
Example 3:
It can be seen from the observation that the function has no domain and no limit at x=2, and none of the three conditions hold.
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chapter10 What is derivative, change is the last word
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The derivative is actually very simple. In a word, it is the "slope"
As shown in the figure, suppose you are walking up the hill with an anesthetized sheep on your back. The coordinates at the foot of the hill are the origin. When you walk up the hill from the foot of the hill, your x and y coordinates will follow your movement at the same time. And change, in fact, is increasing. If H(x) is the height of the hillside at point x, the curve formed by the points of the function of y=f(x) is the outline of the hillside.
Since you care about the steepness of the hillside, and the derivative of h(x) is exactly how steep the hillside is at point x, we denote it by h`(x).
If, h`(10) = 1/6, this means that after walking 10 feet in the x direction, the steepness of the new position you reach is equal to 1/6. The so-called steepness of 1/6 reflects that for every 1 foot you move horizontally, you must move 2 inches vertically (= slope*x = 1/6 foot = 1/6*12=2 inches)
h`(30) = -2, which means that when x=30, for every 1 foot of lateral movement of the ground under the foot, it will move -2 feet vertically. That means you are going downhill.
The derivative f`(x) of the function y=f(x) represents the rate of change of the function. If the derivative is a large positive value, it means that the function is increasing rapidly; if it is a fairly small positive value, it means that the function is also increasing, but only slowly. If the derivative is a negative value, it means that the function is decreasing. If the derivative is equal to 0, it means that the function is neither increasing nor decreasing at least at this instant, maintaining the level.