This question might be related to the question "Is the space we live in continuous (as mathematically defined) or quantized?" but asking over the edge: Are we able to tell? As my understatement of all senses are discrete, as they are detectors based on electrical signals, I wonder if our representation of the world is a natural limit to understand continuity itself? Or posed in another way: can we explain or understand something, that we cannot experience?
Avoiding microphysics, let's say that a wall is a continuous surface. Let's also take for granted your suggestion that our senses are discrete, in that a neuron either fires or does not when stimulated by something like touching a wall.
Now, your problem is that a wall is continuous, while our experience of it is only ever a series of on-or-off neuron firings when we encounter it. Well, we constantly sample things, and then generalize from the samples to wholes. I see one side of a tomato from one angle and I make an "inductive inference" to there being a full tomato in front of me, with many sides, filling in what I have not experienced by reasoning that there is probably a full tomato there. Similarly, if you touch a wall at number of different points, you might infer that it is continuous. It is an "inductive" inference (rather than deductive), because you might be wrong. You are concluding something that goes beyond the evidence you have.
So, we arrive at your next problem: when we claim that a wall is continuous, but have only ever experienced parts of it, do we know what that means? Well, this is a problem only if our understanding of continuity could only come through such experiences. But we can also understand it by thinking about Euclid's definition of a line: a continuous series of points. We can understand that without ever having any experiences at all. And we use that concept to make sense of the wall being continuous, even though we know about it just based on a few points.
Adding: Certainly serious questions remain about concepts and their relationship to our experiences!