If you adapt a file, you have to satisfy the licensing requirements for that file, whatever your changes and whatever the licensing requirements. If the module only specifies Copyright ...
and BSD
, I'd start by seeking clarification from the original author, since there are different versions of the BSD license (two-clause, three-clause and the obsolete four-clause): ideally you want to know exactly which one applies. While you're at it, ask the original author to specify a complete copyright line...
In your first case, I believe the notion of derived work involves some measure of creativity, so deleting a function wouldn't produce a derived work — but you still need to follow the original licensing requirements. (You can look at this this the other way round: you're extracting the BSD-licensed first function only.)
In your second case, you're creating a derived work.
In both cases, since the BSD license allows you to modify the code and redistribute the modified copies, you can perform the changes you envision; in the second case, this makes you an additional copyright holder on the resulting work, so it follows that you should add a corresponding copyright line.
The result should look like
// Copyright © original year, original author
// Copyright © 2016, Paolo Marini
// All rights reserved.
//
// Redistribution and use in source and binary forms, with or without
// modification, are permitted provided that the following conditions are
// met:
//
// 1. Redistributions of source code must retain the above copyright
// notice, this list of conditions and the following disclaimer.
//
// 2. Redistributions in binary form must reproduce the above copyright
// notice, this list of conditions and the following disclaimer in the
// documentation and/or other materials provided with the distribution.
//
// THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
// "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
// LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR
// A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT
// HOLDER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
// SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT
// LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
// DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
// THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
// (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
// OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
(assuming the two-clause form).