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I am reading signal processing first by Mcclellan. In chapter 3, I came across the term "discontinous" as shown underlined in attached photo.

Apparently "discontinuous" means having a gap/break but in attached photo we see square wave is mentioned as discontinuous while triangular wave is mentioned as continuous, although there isn't any apparent gap/break in both graphs.

What is the reason for this difference? In context of dsp?

enter image description here

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  • $\begingroup$ as said below your last quesion on "discontinuous signals", the term is mathematically well-defined, and I linked to that definition. I'm not really sure what this near-identical question is about now. $\endgroup$ – Marcus Müller Apr 3 '20 at 17:35
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(Ideal) square waves are often drawn in a misleading way, because the vertical lines don't actually represent a signal value. The square wave actually jumps instantanously between two values, creating a discontinuity.

In other words, a function that is defined like this: $$f(t) = \begin{cases} 0\,\,\,t<0\\1\,\,\,t>=0\end{cases}$$ is discontinuous. Even if you draw a vertical line from 0 to 1 at $t=0$, the vertical line does not actually represent the function: $f(t)$ is never 0.5, or 0.3, or anything besides 0 and 1. Mathematically, the function has a different value when $t$ approaches 0 from the left ($t$ increasing from $-\infty$ to 0) than when approached from the right ($t$ decreasing from $\infty$ to 0).

The triangular wave does not suffer from this problem (but note that its derivative is discontinuous). Mathematically, whether you approach the peak from the left or from the right, you reach the same value.

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  • $\begingroup$ So in a nutshell you mean, to be continous a graph of signal must take or pass through all values between two levels(1 and 0) .? Graph cannot change immediately from 0 to 1 or vice versa for continuous signal? $\endgroup$ – Man Apr 4 '20 at 6:58
  • $\begingroup$ @Man That's right. $\endgroup$ – MBaz Apr 4 '20 at 14:43
  • $\begingroup$ One more concept, if you can kindly guide me comfortably?what is the effect of sudden/abrupt change in amplitude (such as in unit step and square wave) on frequency spectrum? $\endgroup$ – Man Apr 4 '20 at 17:25
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    $\begingroup$ Any instantaneous change in amplitude implies that the spectrum extends to $\pm \infty$. $\endgroup$ – MBaz Apr 4 '20 at 18:38
  • $\begingroup$ Spectrum extends only to plus infinity or negative infinity or both? $\endgroup$ – Man Apr 5 '20 at 6:51

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