Step by step
The question
Question: decide if this f(x) is continuous at x = 2.
Continuous at x = 2 means three things are true.
One: f(2) has a value.
Two: the limit there exists.
Three: they are equal.
1 Does f have a value?
Which piece holds x = 2?
The one with x ≥ 2.
So f(2) = 5.
Careful!
Having a value is not enough.
The graph must also arrive at that height from both sides.
Check 1 is true: f(2) is 5.
2 Does the limit exist?
From the left, we use x².
We get 4.
From the right, we use x + 3.
We get 5.
The sides do not agree: there is a jump.
So there is no limit.
Check 2 is false.
So f is not continuous at x = 2.
3 Check
Let's check on the graph.
The first piece comes in from the left.
The second piece starts somewhere else: a jump.
We must lift the pen.
Right!
Let's look back.
First: does f have a value?.
Then: does the limit exist?.
Last: check.