# Continuity at a point

## Continuity at a Point

### Definition:

• Continuity at a Point $c$: A function $f\left(x\right)$ is said to be continuous at a point $c$ in its domain if three conditions are met:
1. $f\left(c\right)$ is defined.
2. The limit of $f\left(x\right)$ as $x$ approaches $c$ exists.
3. The limit of $f\left(x\right)$ as $x$ approaches $c$ is equal to $f\left(c\right)$.

### Example:

Consider the function $f\left(x\right)=\frac{{x}^{2}-1}{x-1}$.

Let's investigate the continuity of $f\left(x\right)$ at $x=1$.

### Step 1: Checking if $f\left(1\right)$ is Defined:

$f\left(1\right)=\frac{{1}^{2}-1}{1-1}=\frac{0}{0}$ (undefined)

### Step 2: Evaluating the Limit ${\mathrm{lim}}_{x\to 1}f\left(x\right)$:

${\mathrm{lim}}_{x\to 1}f\left(x\right)={\mathrm{lim}}_{x\to 1}\frac{{x}^{2}-1}{x-1}$

Using factorization or canceling out the common factor $\left(x-1\right)$:

$={\mathrm{lim}}_{x\to 1}\frac{\left(x+1\right)\left(x-1\right)}{x-1}$

$={\mathrm{lim}}_{x\to 1}\left(x+1\right)=2$

### Step 3: Comparing $f\left(1\right)$ and ${\mathrm{lim}}_{x\to 1}f\left(x\right)$:

$f\left(1\right)$ is undefined (as $\frac{0}{0}$), but ${\mathrm{lim}}_{x\to 1}f\left(x\right)=2$.

Since $f\left(1\right)$ is undefined, and the limit of $f\left(x\right)$ as $x$ approaches 1 exists and is equal to 2, the function is discontinuous at $x=1$.

### Conclusion:

The function $f\left(x\right)=\frac{{x}^{2}-1}{x-1}$ is not continuous at $x=1$ due to the undefined value of $f\left(1\right)$, even though the limit as $x$ approaches 1 exists and is equal to 2