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I am reading a nice book on DSP
The Scientist and Engineer's Guide to Digital Signal Processing
It mentions
In electronics, the mean is commonly called the DC (direct current) value.
How can a signal have no DC component if the mean is the DC component? Surely every signal's values can be averaged?
A signal having a mean-value or DC component of zero is commonly referred to as mean-free or as having no DC component. It does not mean that it cannot be averaged, just that the average comes out as zero.
Might be a little inexact but it is very common.
To put in "Frequency" terms, the value of zero frequency corresponds to "DC" by name and it has nothing to do with Direct current actually. The fourier transform at zero frequency is simply an integration of a signal throughout its existence (dividing by the length of the time interval gives the mean value of the signal). Now if the Integration(in the case of continuous time domain) or the summation(discrete time domain) turns out to be zero, essentially you won't have any zero frequency component, and hence the average will also be zero.
Also you have to be careful with the fact that the transforms exist only if the signals don't blow up. Eg. a signal that's just a constant from $-\infty$ to $+\infty$ will blow up at zero frequency. In real world we deal with time limited signals and the signals that don't blow up so it's not an issue.
I will paste a link of a similar question asked in dsp stack exchange a month back. It clearly explains why X(0) is called the dc component and why X(0) is the summation of all the samples x[n] of the signal.
Now for your question on how the signal can have no dc component or mean, you can make the signal mean free by subtracting the average of the signal samples from every time domain sample thereby making the signal mean free both in the time domain and frequency domain. Professor Dilip Sarwate has written a clear comment about the same in the comments section of the link i posted.
