Is a function continuous at the point where it ends abruptly?

Question

Is a function continuous at the point where it ends abruptly?

A function $f(x)$ to said to be continuous at a point $a$ iff:

1) $f(a)$ is defined,

2)$\lim\limits_{x \to a} f(x)$ exists, and

3)$\lim\limits_{x \to a} f(x)=f(a)$

At point $P_3$:

1)Left-hand limit exists, which is equal to 1.

2)The function is defined at $P_3$, which is also equal to 1.

Therefore

$$\lim\limits_{x \to P_3^-} f(x)=f(P_3)$$

So, can it be inferred that the function is continuous at $P_3$ or the rhight hand limit should also exist for the function to be continious at $P_3$?

Answer 1

You can't tell for sure if f(x) is continuous at P3. You can assume it is, but It is only safe to say that between p0 and p3 the function is continuous. If you continue with the graph past P3, you could find it makes an abrupt point at P3. In that case, f(x) is not continuous at P3.

Answer 2

Yes, the function is continuous at both endpoints.

Since the function is only defined on $[0,1]$, the domain of your function is $[0,1]$ and the function is continuous at the endpoints as well as the interior points.

Note that in the definition of $$\lim _{x\to a } f(x)$$ we consider $ x\in D $ where $D$ is the domain of the function.