Why are binary signals drawn continuous in time despite being a digital signal
I can't quite make out where your confusion is but perhaps the following will help.


  A digital signal is a signal that is discrete in time and quantized in amplitude. 


I think that all we're saying here is that the digital signal has some fixed value for a period of time. In your example that's the sample time.


  A binary signal is a Digital signal.


True (but with a small 'd' as 'digital' is not a proper noun).


  A binary signal, also known as a logic signal, is a digital signal with two distinguishable levels.


Agreed.


  But according to the definition, the signal should be discrete in time to be a digital signal. In the plot, a binary signal is drawn as continuous in time. Then how can this be a digital signal according to the definition.




Figure 1. OP's graphic with quantized overlay.

The grey trace in your diagram is the analog signal that is being digitized. The red vertical bars are the result of the analog to digital conversion (ADC). I have superimposed the blue trace which shows what has been stored and what would be produced by a digital to analog conversion (DAC) if you were to play back the stored samples at the same rate that they were recorded. Obviously, this is not a high-fidelity reproduction of the original signal but it could be improved quite a bit with low-pass filtering.

Does that help?



From the comments:


  "But why is the binary signal despite being a digital signal represented as continous in time?" Are you asking why there are horizontal lines on your first diagram? [Transistor]. (OP answered "Yes".)


The horizontal line tells you that the value is unchanging for that period of time. Your first, binary timing diagram might be the output of a serial connection. It's low for one clock cycle then high for one clock cycle, etc. It's what you would see if you monitored the signal on an oscilloscope.

Your second diagram is showing the digitisation of an analog signal. The samples are taken at discrete intervals and the measured values are the red dots. On playback through the DAC you will get the continuous 'curve' that I've shown that in blue on Figure 1. The grey is what you would see when recording. The blue is what you see on playback. The red dots are the values stored or transmitted in the digital circuitry.