Yes. There have been a number of studies to prove it. First off when you sleep you body produces an enzyme that slows your metabolism to a resting rate. The only way the body knows to stop making this enzyme is for you to eat. Once you eat you body starts kicking up to speed, not only allowing you body to function properly, but also your brain.
This body reacts to the enzyme in such a strong way it actually is known to increase the risk for heart attack. There is a higher percentage of people having heart attacks in the morning before eating than any other time of day.
If you eat a breakfast that contains protein it actually improves brain function. I am not saying you need to eat a big breakfast, something small will do. You can boil eggs up ahead of time and grab one or two while heading out the door. A container of yogurt or even a slice of toast with a slice of cheese on it will do.