# Functions, Limits, Continuity and Differentiability

An equation will be a **function** if, for any $x$ in the domain of the equation (the domain is all the $x$’s that can be plugged into the equation), the equation will yield exactly one value of $y$ when we evaluate the equation at a specific $x$.

**Single variable** calculus deals with functions of one variable, **multivariable** calculus deals with functions of multiple variables.

### Limits

A limit is the value **f(a)** that a function **f(x)** approaches as that function’s inputs** 'x'** get closer and closer to some number **'a'**. The idea of a limit is the basis of all calculus.

In the following example, the limit of a function $g(x)$ as the input $x$approaches 2 is a value 4.

### Continuity

A function f(x) is continuous on a set if the graph of f is a connected curve without any jumps, gaps, or holes.

Continuity can be defined using the concept of limits where a continuous function will satisfy the following equation:

### Differentiability

A function is **differentiable **at a point if it has a **derivative **there.
Note: A differentiable function must be **continuous**

The function **f** is **differentiable **at **x **if the derivative exists:

#### Test for differentiablity:

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