What is the Value of a for Continuity in This Function?

[Torshi]

Finding the value of "a"

Homework Statement

Determine the value of "a" such that the function is continuous on the whole real line. You must clearly demonstrate why your choice of satisfies the definition of continuity

Homework Equations

f(x) = { (x+a), x≠a and 8, x=a

The Attempt at a Solution

x+a = 8
a+a = 8
2a=8
a=4

Then from there what do I do? Plug 4 into the first function?
The 8, x=a --> is that undefined or limit does not exist?

 

[mfb]

Then from there what do I do? Plug 4 into the first function?

Right

And show that the limit for x->4 is the same as the value at x=4. That can be done in 1-2 lines.

 

[Torshi]

Right

And show that the limit for x->4 is the same as the value at x=4. That can be done in 1-2 lines.

x+a = 8
x+4 = 8
x=4?

 

[mfb]

Those lines might need an explanation, but the general idea looks good.

 

[Torshi]

Those lines might need an explanation, but the general idea looks good.

That's what I don't understand.

I don't know how to explain the reasoning
Because for the first function x≠a and for 8 x=a which makes since in regards to 4+4 = 8 since a=4 and x=a

 

[Mark44]

f(x) = x+a, if x≠a and 8, if x=a

For f to be continuous at x = a, it must be true that f(a) = $\lim_{x \to a} f(x)$

So we must have $\lim_{x \to a} f(x) = 8$
$\lim_{x \to a} f(x) = \lim_{x \to a} (x + a) = 2a$

Hence 2a = 8, or a = 4.