Is a function with inverse function necessarily continuous

Is a function with inverse function necessarily continuous

Not necessarily. To give you a counter example, f (x) is defined as follows
F (x) = x if x is a rational number
F (x) = - x if x is an irrational number
F (x) is obviously discontinuous
It can be calculated that his inverse function g (x) is f (x) itself
(there is obviously f (f (x)) = x)

Do all functions have inverse functions

Not all functions have inverse functions
In the definition of function, every value in the definition field can only correspond to the y value in a unique value field
Therefore, if the function has an inverse function, if and only if every y value in the range corresponds to a unique x value in the domain
In other words, if different x cannot be mapped to the same function of Y, then there is an inverse function

Are the parity of the original function and the inverse function the same?

No matter even function or odd function, there is inverse function in a monotone interval, and tangent monotonicity is the same as the original function

Finding the inverse function of a function y=(e^x-e^-x)/2 We get e ^ x-e ^ - x = 2Y, multiply the two sides of the equation by e ^ x to get (e ^ x) ^ 2-2ye ^ X-1-1 = 0, which is a quadratic equation about e ^ X, Since the root (y) is y + 2, the root (y) is equal to or greater than 2 My question is, why is e ^ x ≥ 0? In my opinion, taking any value of X, e ^ x will not be equal to 0

≥ is greater than or equal to
That is greater than and equal to one can be established
So it's OK to be ≥ 0 here
Because he mainly said that it would not be less than 0
Which negative solution is omitted

Why the inverse function of monotone function must exist

The definition of function requires that there is a unique dependent variable corresponding to the independent variable in each definition domain. According to our usual representation of X-Y, there is a unique y corresponding to X in each definition domain. However, the function does not require that there is only one X corresponding to different Y values. For example, the function y = x 2

Only monotone functions have inverse functions, Or ask: monotone function must have inverse function,

"Only monotone function has inverse function" is not exact, for example, y = 1 / x, it is not a monotone function, but there is an inverse function (although it is itself). In addition, "monotone function must have inverse function

Are functions with inverse functions necessarily monotonic? Why?

Of course, not necessarily. The counter example: F (x) = x x x in [- 1,0]
10-x at (0,1]
The function has inverse function but not monotone

Must inverse function be monotone function?

not always.
For example, if the piecewise function f (x) = {x, X ≤ 0; - x + 1,0 its inverse function F-1 (x) = {x, X ≤ 0; - x + 1,0 is a sufficient condition for the existence of an inverse function, not a necessary condition

Only monotone functions have inverse functions

In fact, "function Wan = Ding (x) is a single error function" and "only" right = f (x) has inverse function ", but not necessary condition. The following two examples are used to illustrate: Example 1 function, such as = reaches in the definition domain (one OO, 0) u x (O, +)

Finding the inverse function of y = 1 / X

Its inverse function is itself