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O.J.

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I'm taking a mathematics course in probability and stochastic processes and we started covering the CDF (cumulative distribution function) which i understand perfectly and then the PDF (probability density function). The PDF was defined to be the derivative of the CDF. Now, the CDF is defined in terms of cumulative probabilities multiplied by a unit step function. Differentiating that we get the same probabilities multiplied by the imulse (or delta) function. Our professor stated that the sketching of the PDF will produce vertical lines at the end points of each random variable whose magnitude equals the jump between the previous and the current random variable probabilities. However, I am finding it hard to understand that mathematically as the probabilities will be multiplied by the impulse function (say Imp (X - Xi) whose value at Xi is infinite by definition. How is it that we get a finite value at these points when Imp(X-Xi) is supposed to be infinite? can someone please elaborate?