Preheat oven to 350 degrees, and grease a 9 inch cake pan with olive oil, spring form pan, or loaf tin if preferred. You can also add parchment paper on the bottom and grease again.
Mix the flour, baking powder, baking soda, and salt together in a bowl and set aside.
Separate the eggs, and add the egg yolks to a mixing bowl. Set aside the egg whites.
Beat the egg yolks and the granulated sugar for at least 3 minutes, until pale and fluffy. Add in the olive oil, slowly, and keep mixing on a slightly lower speed since the mixture will become more thin with the addition of oil.
Add in the lemon rind, the lemon juice, and the milk. Mix on low speed until incorporated.
Add in the vanilla extract and mix briefly to incorporate again.
Add in dry ingredients spoon by spoon, letting it incorporate before adding more, and beat until the mix thickens again for at least a minute or two. Set aside for a moment.
In a bowl, add the blueberries and blackberries and mix in with the flour, making sure to coat all the fruit completely. Sieve the berries so any excess flour is removed.
Beat the egg whites until they move past foamy and soft peaks are achieved.
Very carefully fold the egg whites into the cake mixture, being sure to integrate without knocking the air out of the egg whites. Add in the flour coated berries and give a few gentle folds of the spatula until the berries are evenly floating throughout the batter. Do not mix too hard or too much, otherwise the fruit will break up and bleed into the cake batter.
Drop batter into the pan and bake for at least 35-40 minutes, checking with a toothpick to make sure no batter appears.
Let cool for 10 minutes, then move to a wire rack to cool completely. Serve with whipped cream, creme fraiche, or simply with confectioners sugar and a more berries. I added fresh flowers for a pretty, but inedible effect. The cake keeps well and can be frozen for a couple weeks to get you to that first note of cool autumn.