For what value of c is the function defined below continuous on (-\infty,\infty)?
f(x) =
{x2−c2,x < 4 ; cx+20,x≥4.

It's clear that for x not equal to 4 this function is continuous. So the only question is what happens at 4.
A function, f, is continuous at x = 4 if 


In notation we write respectively


Now the second of these is easy, because for x > 4, f(x) = cx + 20. Hence limit as x --> 4+ (i.e., from above, from the right) of f(x) is just 4c + 20.

On the other hand, for x < 4, f(x) = x^2 - c^2. Hence 


Thus these two limits, the one from above and below are equal if and only if
 4c + 20 = 16 - c² 
 Or in other words, the limit as x --> 4 of f(x) exists if and only if
 4c + 20 = 16 - c²

That is to say, if c = -2, f(x) is continuous at x = 4. 

Because f is continuous for all over values of x, it now follows that f is continuous for all real nubmers

That they are numbers and related to each other.