Unit Impulse and Unit Step functions.
Switch On the lower panel to multiply f(x-τ) by u or δ.
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I give δ(t) a height of 1 at the peak.
Say I vary the position δ with the parameter t. According to Newton, to represent the figure f(x-τ) by the (convolution) integral f(t)*δ(t), I must maintain a ratio of equality between the areas of the two figures, in the limit as Δt→ 0.
I am making up δ, so I will satisfy this condition by definition:<br>

δ(t) has an area of 1

For example, assume the spike of δ(t) is a narrow rectangle of height h and width Δt. The area is h Δt, which must have the constant value 1. Hence, h = 1/Δt. The rectangle vanishes to a line as Δt → 0, and h → ∞.
I may, then, represent the peak of δ(t) by infinite height and zero width.
I will use an operational definition instead: δ(t) is any function which satisfies
f(t) =f(t)*δ(t)
In other words, δ(t) retrieves the original function.
Voilà. Synthetic, piecewise integration.
To DO: (Clumsy interface, needs work.)
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Worksheets to accompany
The Fourier Transform and its Applications; Prof. Osgood, Stanford University:
http://www.youtube.com/watch?v=gZNm7L96pfY