![f(x)= \left \{ {{x^2-c^2,x \ \textless \ 4} \atop {cx+20},x \geq 4} \right ](https://tex.z-dn.net/?f=f%28x%29%3D%20%5Cleft%20%5C%7B%20%7B%7Bx%5E2-c%5E2%2Cx%20%5C%20%5Ctextless%20%5C%20%204%7D%20%5Catop%20%7Bcx%2B20%7D%2Cx%20%5Cgeq%204%7D%20%5Cright%0A)
It's clear that for x not equal to 4 this function is continuous. So the only question is what happens at 4.
<span>A function, f, is continuous at x = 4 if
</span><span>
![\lim_{x \rightarrow 4} \ f(x) = f(4)](https://tex.z-dn.net/?f=%5Clim_%7Bx%20%5Crightarrow%204%7D%20%5C%20%20f%28x%29%20%3D%20f%284%29)
</span><span>In notation we write respectively
</span>
![\lim_{x \rightarrow 4-} f(x) \ \ \ \text{ and } \ \ \ \lim_{x \rightarrow 4+} f(x)](https://tex.z-dn.net/?f=%5Clim_%7Bx%20%5Crightarrow%204-%7D%20f%28x%29%20%5C%20%5C%20%5C%20%5Ctext%7B%20and%20%7D%20%5C%20%5C%20%5C%20%5Clim_%7Bx%20%5Crightarrow%204%2B%7D%20f%28x%29)
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 <span>4c + 20.
</span>
On the other hand, for x < 4, f(x) = x^2 - c^2. Hence
![\lim_{x \rightarrow 4-} f(x) = \lim_{x \rightarrow 4-} (x^2 - c^2) = 16 - c^2](https://tex.z-dn.net/?f=%5Clim_%7Bx%20%5Crightarrow%204-%7D%20f%28x%29%20%3D%20%5Clim_%7Bx%20%5Crightarrow%204-%7D%20%28x%5E2%20-%20c%5E2%29%20%3D%2016%20-%20c%5E2)
Thus these two limits, the one from above and below are equal if and only if
4c + 20 = 16 - c²<span>
Or in other words, the limit as x --> 4 of f(x) exists if and only if
4c + 20 = 16 - c</span>²
![c^2+4c+4=0 \\(c+2)^2=0 \\c=-2](https://tex.z-dn.net/?f=c%5E2%2B4c%2B4%3D0%0A%5C%5C%28c%2B2%29%5E2%3D0%0A%5C%5Cc%3D-2)
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 ![(-\infty, +\infty)](https://tex.z-dn.net/?f=%28-%5Cinfty%2C%20%2B%5Cinfty%29)