Explanation:

What is an inverse function?
An inverse function is a function that undoes the action of the original function. If the function f takes an input x to an output y, then the inverse function g takes the output y and returns the input x. In other words, g(f(x)) = x for all x in the domain of f, and f(g(y)) = y for all y in the domain of g.

What is the condition for a function to have an inverse?
For a function to have an inverse, it must satisfy the following condition:

- The function must be one-to-one or injective, which means that each element of the range is paired with a unique element of the domain.
- The function must be onto or surjective, which means that each element of the range is paired with at least one element of the domain.

Why does a function need to be strictly monotone and continuous in the domain?
A function that is not one-to-one or onto cannot have an inverse. For example, the function f(x) = x^2 is not one-to-one because both f(2) = 4 and f(-2) = 4. Similarly, the function f(x) = sin(x) is not onto because there is no x for which f(x) = 2.

On the other hand, a function that is one-to-one and onto may still not have an inverse if it is not strictly monotone and continuous. A function is strictly monotone if it always increases or always decreases on its domain. If a function is not strictly monotone, then it may have two different inputs that map to the same output, which violates the one-to-one condition.

A function is continuous if there are no sudden jumps or breaks in its graph. If a function is not continuous, then it may have gaps or holes in its graph, which makes it difficult to define an inverse function.

Therefore, a function must be strictly monotone and continuous in its domain to have an inverse function. This condition ensures that each element of the range is paired with a unique element of the domain, and that the inverse function can be defined without any gaps or jumps.