Fourier Transform and Delta Function I am very new to Fourier analysis, but I understand that through the use of the Fourier transform a signal in the time domain is displayed in the frequency domain, where frequency values are normally displayed along the x-axis, and amplitude is displayed along the y-axis.  However, at one point in the textbook I am using, the following is stated:

Let us assume that we have the function $f(t) = \cos(\omega_0 t)$.  The spectrum then consists of two delta-functions

$$F(\omega) = \pi \delta(\omega - \omega_0) + \pi \delta(\omega + \omega_0)$$

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This confuses me.  When we have $f(t) = \cos(\omega_0 t)$, then I would assume that the Fourier transform should yield an amplitude of $1$ at $\omega = \omega_0$ and $0$ elsewhere. But the delta function is defined as:

$$\delta(\omega - \omega_0) = \left\{ \begin{array}{1 1} \infty &amp; \quad \omega = \omega_0 \\ 0 &amp; \quad \omega \neq \omega_0 \end{array} \right.$$

So wouldn't this give an infinite value at $\omega = \omega_0$? 

If anyone can explain the intuition behind the statement in my textbook, then I would be very grateful!
The textbook is right. A sine wave in the time domain has infinite energy since it continues over an infinite amount of time. When you transform into the Frequency domain all this energy is concentrated on a single (or two) frequency. Hence the value there is indeed infinite. 

Of course these are all theoretical considerations. In the real world, ideal sine waves do not exist since they all have a beginning and an end.