# Continuity at a point

## Continuity at a Point

### Definition:

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

### Example:

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

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

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

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

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

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

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

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

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

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

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

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

### Conclusion:

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