Is the given language DCFL?
You can construct a DPDA for your language as follows. When reading $a$s, the machine pushes $A$s onto the stack. When reading $b$s, the machine pops them, noting to itself if there are more $b$s than $a$s. When it reads the first $c$, it can use the top-of-stack to tell if the number of $a$ was equal to the number of $b$s; if so, it transfers control to a DFA for $(cc)^*$, and otherwise it transfers control to a DFA for $c^*$.