I do have restart, along with gc() at the top of the worksheet.  At this point, the results are consistently wrong.  I ran my worksheet and had it output the equations that were being plotted.  I then copied these equations to a new worksheet and plotted them to attempt to reproduce the problem.  However, in that new worksheet, the plots were created correctly.  I attached that file, if it helps any.  I have Digits declared at the beginning as 32, which is the same as my main file.  The second and third plots in that file are the ones that are the problems.  I would attach the main file, but unfortunately, for confidentiality reasons, I can't do that.  In any case, I have been writing this Maple worksheet in conjunction with a Matlab/MuPAD program hoping to get identical results to confirm that I have not made a mistake.  I now have a Matlab/MuPAD program that works very nicely, and am quickly coming to the conclusion that it is unnecessary to validate my results with Maple.  To say the least, I have not been very impressed with Maple's programming capabilities.  It's unintuitive and seems, at least at this point, to be buggy.

Example.mw

I do have restart, along with gc() at the top of the worksheet.  At this point, the results are consistently wrong.  I ran my worksheet and had it output the equations that were being plotted.  I then copied these equations to a new worksheet and plotted them to attempt to reproduce the problem.  However, in that new worksheet, the plots were created correctly.  I attached that file, if it helps any.  I have Digits declared at the beginning as 32, which is the same as my main file.  The second and third plots in that file are the ones that are the problems.  I would attach the main file, but unfortunately, for confidentiality reasons, I can't do that.  In any case, I have been writing this Maple worksheet in conjunction with a Matlab/MuPAD program hoping to get identical results to confirm that I have not made a mistake.  I now have a Matlab/MuPAD program that works very nicely, and am quickly coming to the conclusion that it is unnecessary to validate my results with Maple.  To say the least, I have not been very impressed with Maple's programming capabilities.  It's unintuitive and seems, at least at this point, to be buggy.

Example.mw

@Robert Israel 

Entering the command

at least for the time being gives

Also, entering

oddly enough produces the slightly different result of

Who knows, maybe tomorrow the Maple software will be looking kindly on me and plot it the way I want it to, if I ask nicely enough.  :)

So I guess this shouldn't be too much of a surprise to me with as unreliable as Maple seems to be, but I just replotted the function below and got the same result as I got before this morning.  I did not change anything anywhere in the whole worksheet.  How in the world could this be happening?

gives

At this point, only 7 out of the 16 plots I am trying to make are plotting correctly.  They were, at one time, plotting correctly, but now they are not.  It's not even like some setting outside of the call to the plot command could be effecting it (if that's even possible), since nothing was changed in the entire worksheet from this morning till now.  I suppose it's probably something stupid and obvious that I'm missing, but at this point, it's got me a little frustrated, to say the least.

Anyhow, thanks again for your help.

@Robert Israel 

Okay so Maple sometimes really confuses me.  Last night I tried what you said and it produced exactly the same plot as before.  This morning, I re-ran the worksheet and tried plotting again and this time one (though not both) woprked correctly.  Now, entering

produces

You will note that the input is exactly the same as it was before, though the output is different.  Though this is less of a concern, it puzzles me as to how this could happen.  In any case, the other plot (of the governing DE) still produces the same result.  Entering

produces

I just wish Maple were as intuitive to use as Matlab.

Basically, to summarize my question: How do I get the second and fourth plots to look similar to the first and third plots?

@Markiyan Hirnyk 

Well, I'm not sure because the expressions that I am plotting are very large and complicated.  I will do my best though.

The solution I am trying to obtain is the solution to a contact problem from elasticity.  For example, I am plotting the governing differential equations which I desire to evaluate to zero within working precision.  In the region of contact, this works fine, but in the non-contact region it doesn't.

gives

but

gives

The second plot should look something like the first, but Maple decides to plot it on a scale from -infinity to +infinity.  I don't know if that helps any, but a similar thing happens when I try to plot the displacements resulting from the contact.  Entering

gives

but

gives

Note that the only difference between the last two plots was the change in the lower limit for the range of θ from slightly below the defined range in the discontinuous function to right at it. Also (@PatrickT), note that the "view" option has been defined, but did not prevent Maple from plotting the function on an infinite scale.  I hope all that isn't too overwhelming.

Thanks for your help.