Given a function h(t) specified for all t, we define the continuous Fourier transform (FT) of h(t) as
The inverse transform may then be shown to be
You should note carefully the sign convention in these definitions,
and the presence of the factor 
in the exponent. Other
definitions of the Fourier transform may have a different sign
convention and may use w, the angular frequency, rather
than f.
If t is regarded as the time in seconds, then f is the frequency in cycles per second. Of course, FT pairs apply not only to the time/frequency double, but also to many other pairs. For example, in optics and antenna theory, position in the electromagnetic field distribution across an aperture produces an FT pair with the angular distribution of the far-field electromagnetic field.
The continuous Fourier Transform defined here can be related to the Fourier
series that you have studied in
Chapter 1 by allowing the period of the function to
trend to infinity. The fundamental then tends to zero frequency, and the
overtones become more and more closely spaced. The summation of harmonics
is replaced by the corresponding integral over frequency.