If you write a test for every single function on your system, and your system keeps changing, how will you know when a function is not necessary anymore?
Writing a test for every single function on your system may come from the "100% Coverage Syndrome", which afflicts some managers, thinking that the only way to be completely sure your system is "bug free" is to write tests for every single line of code, till you reach the magical "100% coverage" in all the tests.
I do believe you can reach 100% coverage, as long as you're willing to delete your code.
(Cue the universal grasps here.)
But how do you know which pieces of code can be deleted?
When I mentioned integration tests, I mentioned how much more sense it made to me reading them instead of the "unit" tests, because they were describing exactly how the system would operate in normal (and some abnormal) conditions. If you write tests that go through the system, assuming it is a black box with an input point and an output, and you can get tests for all the normal cases -- and some "abnormal", like when things go wrong -- then you know that, if you run those tests and they mark some lines as "not tested", it's because you don't need them.
"But Julio, you're forgetting the error control!" I do agree, specially when you're talking with project owners or some other expert, that people will forget to tell you what to do in case of things going wrong -- say, the user typing their name in the age field -- but you can see those and you know that you need error control so you can add the error control and describe the situation where that error control would trigger.
If, on the other hand, you write a test for every function, when you do a short/simple check, you'll find that the function is still being used in the system by the tests, not actually, "value to the user" code. Sure, you can use your IDE to go back and forth between code and test and see if it points a use beyond the test, but it won't do it for yourself.
There is one other weird thing about using integration tests for error controls: Sometimes, you can't reach the control statement. It's true! I did wrote control checks for every function once but, when running in the integration tests, there was no way to produce an input at the input layer of the system that would reach the error control in that function 'cause the other functions, which would run before the one I was trying to test, would catch the error before it. If that's a design problem or not -- it probably was -- it's a different discussion, but the fact is that that function didn't need error control, something that I wouldn't see if I wrote test specifically for it, but it was clear in an integration test run.