Check if array is monotonic i.e either monotone increasing or monotone decreasing
static
IsMonotonic(...) does not need an instance of the MonotonicArray class to function, therefore it should be static.
Consistency
You special case an array of length 1 as monotonic.  Is it really?  It is neither increasing nor decreasing.
What about IsMonotonic(new int[]{1, 1, 1, 1})?  Seems to me that should be true, but it will return false.  Definitely should be added as a test case.  And if it should return true, then ...
Optimization
... checking for length 1 is too restrictive.  Any length 2 array will always be monotonic as well.  Perhaps:
    if (numbers.length == 1) {
        return true;
    }

should be:
    if (numbers.length &lt;= 2) {
        return true;
    }

Looping
This is ugly.  Will Java optimize the numbers.length - 1 calculation as a constant?
    for (int index = 0; index &lt; numbers.length - 1; index++) {

        if (numbers[index + 1] == numbers[index]){
            continue;
        }
        ...

It may be better to use Java's enhanced for loop to extract numbers, and rely on monotonic behaviour allowing equality to handle the first element:
    int current = numbers[0];
    for(int value : numbers) {
        if (value != current) {
           if (value &lt; current) {
              ...
           } else {
              ...
           }
           current = value;
        }
    }