Given a sequence of positive integers A and an integer T, return true if a continuous sequence of A sums up to exactly T
Approach and complexity

It is a pleasure to tell you this: Excellent!

Your approach is precisely what I had in mind when I read the challenge description (before looking at your code).

Unit tests

Excellent that you have unit tests! Definitely helps in checking if it's possible to clean up your method. My only suggestion would be to make them Parameterized, which should be a lot more helpful if one or more of them fails.

Some simplifications

I was slightly confused by this code at first:

if (array[i] &gt; k) {
    i += 2;
    start = i;
    sum = 0;
    continue;
}


Then I noticed that it checked if the value at the current position in the array was more than the target k. Then I figured: "Why use this as a special case? The rest of the code should be able to handle this." And it does. Removing this part of the code doesn't change a thing on your unit tests either. While I can understand why you are doing this (it probably does help a bit with performance in some cases), I don't think it's worth it. Don't add unnecessary special-cases.



Next thing I noticed is that sum &lt; k will always be true. Therefore, that if can be removed and be simply:

sum += array[i];
i++;


And now, speaking of i++, we can transform this for-loop into a traditional for loop, by adding both the initialization and the iteration to it:

for (int i = 0; i &lt; array.length; i++) {


This makes it:

static boolean findSum(final int[] array, final int k) {
    int start = 0;
    int sum = 0;
    for (int i = 0; i &lt; array.length; i++) {
        sum += array[i];
        while (sum &gt; k) {
            sum -= array[start];
            start++;
        }
        if (sum == k) {
            return true;
        }
    }
    return false;
}


Nitpicks

k is a bad variable name. target or targetSum would be better.

I am not a big fan of int start = i; in your original code. start is not dependent on i at all. They start at the same value, sure, but it's more clear to set start = 0;

Your method lacks an accessibility modifier, you probably want to make it a public method.

Overall

Well done!