Is this a Context Free Language? I got this question on my final exam: Is the following language context-free?

$$ L = \{w\bar w^R \mid  w\in \{0,1\}^*   \}$$

Notation: 
The string $\bar w$  is obtained from $w$ by replacing all 0s with 1's and all 1's with 0's.  The string $\bar w^R$ is $\bar w$ in reverse order.




  I've thought about it being a context-free language, but I notice that
  when you pump the string in the middle, the string will still be in the
  language. (using pumping lemma)


I think it's context free. 
This is the context free grammar:

$$
S \to 0S1 \mid  1S0 \mid \varepsilon $$
(It's basically a palindrome, but both sides are exact opposites.)
A Context free grammer is one in which LHS of atleast one of the productions is free from any terminal symbols (context). 
According to your production rule,it is clearly CFG.