I saw this link but I'm not asking for a performance degradation for code using "extern". I mean without "extern", is there "context switching" when using C library in C++? Are there any problems when using pure C (not class-wrapped) functions in C++ application?
Both C and C++ are programming language specifications (written in English, see e.g. n1570 for the specification of C11) and do not speak about performance (but about behavior of the program, i.e. about semantics).
However, you are likely to use a compiler such as GCC or Clang which don't bring any performance penalty, because it builds the same kind of intermediate internal representation (e.g. GIMPLE for GCC, and LLVM for Clang) for both C and C++ languages, and because C and C++ code use compatible ABIs and calling conventions.
In practice extern "C" don't change any calling convention but disables name mangling. However, its exact influence on the compiler is specific to that compiler. It might (or not) disable inlining (but consider -flto for link-time-optimization in GCC).
Some C compilers (e.g. tinycc) produce code with poor performance. Even GCC or Clang, when used with -O0 or without explicitly enabling optimization (e.g. by passing -O1 or -O2 etc...) might produce slow code (and optimizations are by default disabled with it).
BTW, C++ was designed to be interoperable with C (and that strong constraint explains most of the deficiencies of C++).
In some cases, genuine C++ code might be slightly faster than corresponding genuine C code. For example, to sort an array of numbers, you'll use std::array and std::sort in genuine C++, and the compare operations in the sort are likely to get inlined. With C code, you'll just use qsort and each compare goes through an indirect function call (because the compiler is not inlining qsort, even if in theory it could...).
In some other cases, genuine C++ code might be slightly slower; for example, several (but not all) implementations of ::operator new are simply calling malloc (then checking against failure) but are not inlined.
In practice, there is no penalty in calling C code from C++ code, or C++ code from C code, since the calling conventions are compatible.
The C longjmp facility is probably faster than throwing C++ exceptions, but they don't have the same semantics (see stack unwinding) and longjmp doesn't mix well accross C++ code.
If you care that much about performance, write (in genuine C and in genuine C++) twice your code and benchmark. You are likely to observe a small change (a few percents at most) between C and C++, so I would not bother at all (and your performance concerns are practically unjustified).
Context switch is a concept related to operating system and multitasking and happens on processes running machine code executable during preemption. How that executable is obtained (from a C compiler, from a C++ compiler, from a Go compiler, from an SBCL compiler, or being an interpreter of some other language like Perl or bytecode Python) is totally irrelevant (since a context switch can happen at any machine instruction, during interrupts). Read some books like Operating Systems: Three Eeasy Pieces.