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

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 $P_3$?

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.
• Mohammad. Nice, "we consider $x \in D_f$"! – Peter Szilas Apr 10 '18 at 8:55