Error caused by Scope or Function Generator?
Ancient but superb oscilloscope!


Oscilloscope probe may need adjusting but othewise the scope is well capable of handling this signal. 
The waveform shows what would probably be expected from a digital generator running near the top end of its frequency range. 


Try adjusting probe used to give a correct square wave response. You will find a small adjustment screw on probe, accessed with an about 1/8" staight bradedscrew driver. Turn slowly, do not force - less than 1 turn total travel. Observe results on this waveform and adjust for best rise time shape and then use calibration output waveform on scope. This is at lower middle of picture. 

The waveform does appear to have genuine steps in it of about 50 ns duration, which is about what you'd expect from a piece of digital equipment clocked at about 5 Mhz. Note that this is not necessarily "bad" and that modern equipment will no necessarly be substantially better if the spec is also not better. Depending what you are doing this may be very acceptable. As long as you know the limitations of your equipment you can often accept using them near the limits and make suitable allowances. 

Report back.



ADDED:

You have substantially changed your question - and also made an observation that needs commenting on: 

The input waveform is definitely not a sawtooth. It appears to be a nominal square wave wity digital steps in it and some slew rate limiting somewhere. 

The addition of the inverter changes the question.
The inverter output is designed to be high or low and to slew rapidly. When the stepped input signal is applied the output stays low as the signal increase in level until a trigger level is reached and then the output goes low "rapidly". This eliminates the stepped effect.

The signal shown is not a perfect square wave. It has non vertical transitions and some "stuff happening" at the transition points. IF that scope has a bandwidth limiting feature )it may have) there will be a button or switch to switch it on and off. Check - if found, ensure it is off.



Oscilloscope probe calibration:

From this very good oscilloscope calibration page