Just a little confused...

I know that this is so rudimentary. However, my professor made this comment and I am really struggling trying to figure it out myself.

Let's say that I have a signal in the time domain that looks like this:

f(t) _____ | __/ _ | _/
| _/ |

---

| t-> | | |

Now, let's say that we sample this signal with the ideal sampling function (comb) delta(t + nTs) [-inf < n < +inf]:

delta(t + nTs)

                      |
                      |
                      |
  ^    ^    ^     ^    ^    ^    ^

||||||__| | t-> | |

Now, the sampled result is delta(t + nTs) X f(t), which results in:

delta(t + nTs) X f(t) ==> x(t)

| . . . | . . . | . . . |

---

| u-> | | |

Okay, so I know a few things. I know that f(t) has a maximum frequency and is therefore band limited in the frequency domain. It is not, however, time limited since both the sampling function and f(t) exist for all t (given the constraints of the sampling function).

So, here is the part that confused me: Even though x(t) is not continuous in the time domain, it is continuous in the frequency domain, where F(w) is:

F(w) | | / \ / |\ /
/ \ / | \ / \ /_../|____/___ | |<-1/Ts->| w-> | |

Is F(w) continuous only because F(w) = 0 over the bandwidth 1/Ts?

I think I just answered my own question. Thanks r/dsp!