Why were Courbet’s paintings radical for their time? (Please include examples from his works.)

Oh, there can be no doubt of the controversial nature of
Courbet's work on two fronts:  depictions of rural life and of the erotic.  Let's
discuss each of them in turn.

At the time in France,
depicting the working class in artwork was considered inappropriate.  Highlighting the
everyday activities of those who would be considered "poor" even more so! Probably the
best example of this is Courbet's The Burial at Ornans which
depicts a common peasant burial in France.  In addition to the fact that the lower class
is represented, there is no real sadness is apparent in the work that focuses only on
caricatures of the subjects.  As a result, many critics called it simply "ugly."  As one
art historian admitted:

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In Paris the Burial was
judged as a work that had thrust itself into the grand tradition of history painting,
like an upstart in dirty boots crashing a genteel party, and in terms of that tradition
it was of course found
wanting.

But there can't be a
discussion about Courbet's controversial nature without talking about his more erotic
works.  A painting (Sleep) of two women in bed together, buck
naked, was simply unheard of in the art world at the time.  Again, one must discuss here
Courbet's most significant work:  The Origin of the World which
very realistically depicts a graphic view of female genitalia (both ironic and a tad
humorous).

As a result, Courbet doesn't fit well into any
of the artistic schools in France at the time (i.e. he is considered neither part of the
Romantic nor part of the Neoclassical movement).  In fact, Courbet himself admitted the
following:

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The Burial at Ornans was in
reality the burial of Romanticism.