Op Amp operating conditions

Like you said, the fact that the two opamp inputs will be nearly equal is a simplification and depends on parameters often not explicitly stated. This is a good question in that it is essential to know the limits of any shortcuts or rules of thumb you use.

As clabacchio already said, one place the assumption are violated is if the opamp output is clipped, or would need to exceed its available range to make the desired signal. Other reasons that make the assumption invalid include:

1. The feedback isn't negative. This may sound stupid, but I've actually shown someone a simple opamp hysteresis circuit in a interview and asked them to draw a plot of output voltage as a function of input voltage. More than one candidate started off saying the opamp will try to keep its two inputs the same, and then dug himself into a deeper hole from there. Needless to say, those were short interviews.

2. The gain isn't sufficient. Note that the rule of keeping the inputs equal assumes infinite gain. Likewise the rule that Gain = -Rf / Rin assumes infinite gain. Normally opamp open loop gains are around 100k or more and we don't ask for more than 100 or maybe 1000 at most from a single stage, so it would seem this is a small issue.

   However, that forgets about the effect of frequency on gain. A 1 MHz opamp may be specified for 100k open loop voltage gain at DC, but if you use it for audio and want to pass 20 kHz, then you only have a open loop gain of 50 worst case. If you set the feedback resistors for a gain of 25, that only leaves 2x headroom at the high end, which will seriously reduce closed loop gain at high frequencies.

3. Slew rate limitation. Even with enough gain and proper feedback, the opamp can only change its output so fast. That is what the slew rate spec is for. The gain*bandwidth product is for small signals. Large amplitude signals may run into slew rate issues. For most opamps, the full swing output signal is rather lower frequency that what the gain*bandwidth product implied.

Positive feedback and negative feedback can be combined to create oscillation (astability). Aside from that, I think you're correct. In the case of the integrator, there's never a step change in the output, so there's never a discontinuity in the capacitor voltage. The only limit is the output voltage range of the op amp.

Assuming an infinite gain with negative feedback implies that the voltages at the two inputs will be equal. Assuming an infinite gain with positive feedback implies that the output will be saturated (unless V+ exactly equals V-).

Even in a real op amp, there's very little current into the input pins (microamps at most). Likewise, the open-loop gain is usually very large at low frequencies (on the order of 10^5 or 10^6 V/V). So the ideal op amp rules are often a pretty good approximation. But all real op amps are limited by their supply voltages, so sometimes that limit is included in an ideal model as well.