The differentiability and differentiability of F(x) are equivalent, but the continuity is not necessarily differentiable (differentiable), and the differentiability must be continuous. (The absolute value of counterexample Y = is continuously non-derivable)

I'm sorry, this example is not invented, but summed up by doing the problem. Many counseling books take this example as a counterexample, and this example is also general. If the degree of x is changed to n (n(n>2)), the derivative of n- 1 is discontinuous. I can't say that this is an extremely rare case, but it must be noted that the derivative is not necessarily continuous. Some topics take advantage of this to set traps, and there is a kind of topic to find the limit. Because of this, we can't use L'H?pital's law (because after L'H?pital's law, the derived derivative function is not necessarily continuous and can't be directly brought in by the function value), but must be defined.

Look at the picture in detail: