You are declaring a local variable, not a class variable. To set an instance variable (attribute), use
class Example(object):
def the_example(self):
self.itsProblem = "problem" # <-- remember the 'self.'
theExample = Example()
theExample.the_example()
print(theExample.itsProblem)
To set a class variable (a.k.a. static member), use
class Example(object):
def the_example(self):
Example.itsProblem = "problem"
# or, type(self).itsProblem = "problem"
# depending what you want to do when the class is derived.