Dirichlet Conditions for convergence of Fourier Transform
Consider an aperiodic signal x(t). Its Fourier Transfrom exists (i.e the Transform integral converges) and 
converges to x(t), except at points of discontinuity provided: 1)
x(t) is absolutely integrable . i.e: 
2) x(t) has only a finite number of extrema in any finite interval.
For example,  does not satisfy this condition in, say (0,1).
3 ) x(t) has only a finite number of discontinuities in any finite interval. For example the following function (the so called Dirichlet's function ) will not satisfy this condition.
These 3 conditions satisfied,
will converge to x(t) at all points of continuity of
x(t). At points of discontinuity of x(t), this integral
converges to the average of the left hand limit and
the right hand limit of x(t) at that point.
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