Constructing a discontinuous voltage waveform as a sum of sine waves would require an infinite number of sine waves. A "perfect" digital signal will switch instantaneously between VSS and VDD (or vice versa); such instantaneous switching events would represent discontinuities in the waveform.
In practice, chips neither produce perfect digital signals on their outputs nor require them on their inputs. Examination of an output signal on a good enough scope will typically reveal that it is a little bit "smoothed off", and most chips will tolerate having input signals which are substantially "smoothed off", provided that they don't linger or bounce around in the region between 1/4 and 3/4 VDD. Indeed, some chips are designed to deliberately smooth off their output waveforms (sometimes by a programmable amount) and/or accept inputs which are smoothed off even to the point of being a bit "mushy".
It's worth noting that while something like a 1 Hz perfect square wave might be expressed as the sum of continuous sine waves ranging from 1 Hz to 1 MHz and beyond, it is very unlikely that any apparatus which is designed for receiving a 1 MHz signal would perceive a 1 Hz square wave as having a continuous 1 MHz component. The 1 Hz square wave would contain, among other things, a 999,999 Hz component whose strength was 1/999,999 of the fundamental, and a 1,000,001 Hz component whose strength was 1/1,000,001 of the fundamental. The apparatus that was trying to receive a "1 MHz" signal would detect those components, and many others, to varying degrees; during each one-second interval, there would be times when they would all be in phase, and times when about half would be in phase and half out. The apparatus would thus perceive a variable amount of "1 MHz" signal--most likely detecting a substantial amount near the moments when the input was switching (because all of the detected waves would be in phase), and a much smaller amount at other times (because the detected waves would have a mixture of phases). A really sharp 1 Hz square wave, driving a strong antenna, would thus not cause continuous interference on a 1 MHz transmission, but rather would yield a 2 Hz "tick tick tick".