Q: Can a discontinuous function be approximated by a fourier series?

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Yes. For example: A square wave has a Fourier series.

yes it can, if you know how to use or have mathematica have a look at this demo http://demonstrations.wolfram.com/ApproximationOfDiscontinuousFunctionsByFourierSeries/

An infinite sum of continuous functions does not have to be continuous. For example, you should be able to construct a Fourier series that converges to a discontinuous function.

Fourier series is the sum of sinusoids representing the given function which has to be analysed whereas discrete fourier transform is a function which we get when summation is done.

A Fourier series is a set of harmonics at frequencies f, 2f, 3f etc. that represents a repetitive function of time that has a period of 1/f. A Fourier transform is a continuous linear function. The spectrum of a signal is the Fourier transform of its waveform. The waveform and spectrum are a Fourier transform pair.

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yes a discontinuous function can be developed in a fourier series

Yes. For example: A square wave has a Fourier series.

Yes, a Fourier series can be used to approximate a function with some discontinuities. This can be proved easily.

yes it can, if you know how to use or have mathematica have a look at this demo http://demonstrations.wolfram.com/ApproximationOfDiscontinuousFunctionsByFourierSeries/

An infinite sum of continuous functions does not have to be continuous. For example, you should be able to construct a Fourier series that converges to a discontinuous function.

Fourier series is the sum of sinusoids representing the given function which has to be analysed whereas discrete fourier transform is a function which we get when summation is done.

no every function cannot be expressed in fourier series... fourier series can b usd only for periodic functions.

sinc^2(w)

A Fourier series is a set of harmonics at frequencies f, 2f, 3f etc. that represents a repetitive function of time that has a period of 1/f. A Fourier transform is a continuous linear function. The spectrum of a signal is the Fourier transform of its waveform. The waveform and spectrum are a Fourier transform pair.

When we do a Fourier transformation of a function we get the primary term which is the fundamental frequency and amplitude of the Fourier series. All the other terms, with higher frequencies and lower amplitudes, are the harmonics.

Consider a periodic function, generally defined by f(x+t) = f(x) for some t. Any periodic function can be written as an infinite sum of sines and cosines. This is called a Fourier series.

Fourier series and the Fourier transform