This much-reprinted article from the Washington Post reports on the Hibi Shimbun, the only newspaper in Ishinomaki, Japan. Hibi Shimun resumed publishing only two days after the earthquake that has devastated Japan and continued publication after the catastrophic tsunami that followed. Incredibly, in the absence of internet or electricity, the first editions post-earthquake were copied out by hand and distributed to local emergency centres. The paper also ceased publishing for two days after the tsunami. According to chief reporter Hiroyuki Takeuchi (and who could argue?), “People who suffer a tragedy like this need food, water and, also, information.”
Electricity has been restored to Ishinomaki, though not internet access, and the newspaper is now printed instead of handwritten.
The article also reports on Kahoku Shimpo, the main newspaper in Sendai, which has continued publishing throughout the disasters and their aftermath.
These would appear to be anomalies in Japan, whose internet has, reportedly, remained largely intact.
Edit: Thanks to glecharles for the link.