很简单的一个道理，比如我定义f(x) = 1/x 在原点之外，然后补充定义f(0) = 0在原点，
这个函数明显就在补充定义的地方是不连续的，所以你需要验证这种不是通过初等函数自然定义的地方（1/x 可以自然定义在x不等于零的任何区域，所以其他地方都是自然定义域，一定连续）
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任何不是自然定义出来的函数（初等函数有限复合），例如用无穷级数定义的函数，或是你说的分段函数，或者我们这里这样的添加部分点定义的函数，都需要验证
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我不认为你需要证明x不等于0的区间，你不可能看到一个函数都要去证明它连续吧。
那我问你sin(cos(e^(1/x^2)))这个函数在x不等于零的时候是不是连续的，你的答案是什么？
明显是连续的，任何初等函数在自然定义域上都是连续的。
x不在原点的情况是它的自然定义域，连续性是显然的，不需要证明，就像你知道sin(cos x)这样的函数必定连续，因为它是初等函数的有限复合，在自然定义域必定连续
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因为我们要求它的导函数，在求导之前至少要确定一下这个函数是连续的
不像e^x, sin x这种自然定义域是全实数轴上的函数，这个函数在原点振荡，所以它并不是显然连续的，你需要证明它的连续性
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这个函数可导，一定连续，但是导函数不一定连续，除非导函数极限存在
Previoualy I tried to claim that the function has continuous derivative at that point and so the derivative must stay negative in its neighborhood, which makes it decreasing immediately.
But later I realized that we couldn’t use this result because this theorem (a derivative is continuous) only holds when the derivative has a limit at that point (导数极限定理）. In our case, we only know thw value of the derivative at that point, but we do not know whether the limit exists. That’s the essential ingredient that renders the first argument invalid.
The second example is more about experience and trying. You want to have something oscillating extremely frequently near a point, and the best choice is something like sin(1/x), which oscillates near the origin. We then need to modify this function to make it havr derivative -1 at the origin. It takes experience (or perhaps mathematical intuition) and many different trials to get a function that works. No one can think out this example in head withiu suffering several failures. (Well, perhaps some genious mind can)
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Updated with more explanation.
Feel free to ask if still confused.
Hi, I am terribly sorry that I messed up question (b) in my first answer.
The theorem I mentioned only holds when the limit of derivative exists at a certain point, but not for any general functions. Here is a counterexample
