php: catch exception and continue execution, is it possible?

Yes but it depends what you want to execute:

E.g.

try {
   a();
   b();
}
catch(Exception $ignored){
}

c();

c() will always be executed. But if a() throws an exception, b() is not executed.

Only put the stuff in to the try block that is depended on each other. E.g. b depends on some result of a it makes no sense to put b after the try-catch block.

Sure, just catch the exception where you want to continue execution...

try
{
    SomeOperation();
}
catch (SomeException $ignored)
{
    // do nothing... php will ignore and continue
    // but maybe use "ignored" as name to silence IDE warnings.  
}

Of course this has the problem of silently dropping what could be a very important error. SomeOperation() may fail causing other subtle, difficult to figure out problems, but you would never know if you silently drop the exception.

Sure:

try {
   throw new Exception('Something bad');
} catch (Exception $e) {
    // Do nothing
}

You might want to go have a read of the PHP documentation on Exceptions.