Why do we define the convolution? [duplicate]

This question already has an answer here:

Why do we define the convolution?

Why is convolution useful?

What is the purpose of the geometry of convolution of two functions in plane? Can we draw the convolution of two functions without compute the integration?

What is idea of definition of convolution?

What is relation between convolution and probibility?

marked as duplicate by Rahul, Math1000, user223391, Leucippus, levapDec 10 '16 at 2:55

• You may find this post to be useful – Omnomnomnom Dec 9 '16 at 17:25
• Was there something in the Wikipedia article that you didn't understand? What would you like clarified? – Thomas Andrews Dec 9 '16 at 17:28

As a special case, the convolution of a function $f(t)$ with the Dirac delta function gives $f(0)$ (the response of the system at $t=0$).
Because of the above, the input signal is thought of as "smearing" the impulse response on the time axis. Mathematically, it means that convolution of a given function $f$ with a smooth function (a "smooth smearer") results in something smoother than $f$. This type of smoothing is extremely useful when you want to use differentials.