In addition to what Grime has said. You will never reach the return statement unless you enter a number greater than seven as the "return" statement is only reached if you reach the "default" case. The "return" statement needs to be outside the switch.
When you read about functions you will see that "int" in the parameter list is not needed.
Unless you need the function this would work just as well:
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#include <iostream>
#include <string>
int main()
{
std::string days[]{ "Sunday", "Monday", "Tuesday", "Wednesday", "Thursday", "Friday", "Saturday" };
int day{};
std::cout << "Day Number = 0 1 2 3 4 5 6 \n Enter Day Number: ";
std::cin >> day;
std::cout << "\n The day is: " << days[day] << std::endl;
return 0;
}