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
// 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).