What are 'Piecewise Functions'?

 A single function that uses different equations depending on the input, like

f(x) = x for x < 0 and f(x) = x^2 for x >0.

Such functions are essential to calculate the left hand and right hand limits of a function at that specific break point, say x = c. 
If the LHL = RHL, we say that Limit Exists at that point.

And If LHL = RHL = f(c), then we say that f(c) is the Limit of the function at 'c' and the function is 'continuous' in the given domian (interval).