Actually, what gets me in trouble all the time (that’s why I’m doing the exam for the 4th time) is the need to get down to basics!
Now, how does this definition of covariance lead to a covariance of zero, if X={100,100, 100, 100c, …}, all same value, as an example? Go over the definition and fit it here in this example. I’ll try: it measures how frequently the largest values of X (which is 100 here) correspond to the largest values of Y, the smallest values of X (which is 100 here) correspond to the smallest values of Y, and everything in between. How does that lead to cov=0? Let us stick to this example, and not repeat the usual cov equation.
X is always the same, so how well do high/low values of X correspond with high/low values of Y? Absolutely not well. So not well, in fact, that they correspond 0, because what happens to Y never has any indication of what will happen to X and vice versa. It’s like asking what the covariance is between a market price and the number 8. What information can you possibly take from that? Zero.