1. Memory backup apparatus for a fault-tolerant computer system having system memories for storing data, a processing element for performing a plurality of data processing tasks and computations, said processing element generating write control signals to cause information to be written into said system memories and task control signals for controlling said system during a context switch, and means for monitoring fault occurrences in said computer system and for generating fault signals, said backup apparatus comprising, a first memory area and a second, physically separate memory area located in said system memories, said memory areas storing duplicate copies of data and subsequent computational results generated by said processing element; a temporary storage area generating a full signal when a predetermined number of storage locations in said memory having been written to; first means responsive to said write signals for writing data from said first memory area into said temporary storage area and for writing computational results produced by said processing element into said temporary storage area; second means responsive to said full signal and to said task control signals for writing selected portions of said temporary storage area into said first memory area when said temporary storage area is full or when said task control signals indicate a context switch is requested, said second means producing a completion signal when said writing has been completed; third means responsive to said completion signal and to said fault signals for writing said selected portions of said temporary storage area into said second memory area when the writing of data from said temporary storage area into said first memory area has been completed without a fault condition being detected; a backup status register and means connected to said second and said third writing means for updating backup status information stored in said backup status register, said backup status information identifying memory areas which have been written from said temporary storage area, wherein said second writing means generates a start signal at the beginning of a storage operation to said first memory area and said updating means is responsive to said start signal for updating said backup status register and is responsive to said completion signal for updating said backup status register; means responsive to said backup status information for restarting a data processing task utilizing said initial data stored in said first memory area if a system failure has occurred before the beginning of a storage operation from said temporary storage area to said first memory area; and means responsive to said backup status information for writing the contents of said second memory area into said first memory area and restarting the data processing task using data stored in said second memory area if a system failure has occurred after the beginning of a storage operation from said temporary memory to said first memory area but before said storage operation has been completed.