The description of frequency on Wikipedia is:
Frequency is the number of occurrences of a repeating event per unit of time
Now, when talking about images, I suppose we're talking about spatial frequency, so it would be per unit of space, instead. But notice the key words repeating event.
So in my understanding, frequency is a property of something that repeats. But edges in an image don't repeat(or I guess you could say they repeat, just very rarely, so we can think of the other repetitions being outside the image). Then why do we say edges are high frequency?
I have read somewhere that edges are regions of the image where the intensity changes rapidly, so that's why it's high frequency. The way I understand it is: High frequency implies rapid change. Not an equivalence relation but an implication. I don't think rapid change always implies high frequency because that rapid change might not repeat periodically.
Where is the fault in my thinking and why are edges high frequency components of an image?
Another thing I've come across recently is the notion of instantaneous frequency. Are edges regions of high instantaneous frequency of an image?
They mean high-frequency in the sense of the Fourier transform. Sharp variations (discontinuities) induce significant components in the high frequencies of the spectrum.
https://en.wikipedia.org/wiki/Fourier_transform
Contrary to the Fourier series, the Fourier transform deals with non-periodical signals. This is a very broad topic, of a fundamental importance in the theory of signal processing.
Intuitively, you need high frequencies to achieve a large slope for a fixed amplitude.