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.

$$Single\hspace{0.1cm}variable\hspace{0.1cm}function:\hspace{0.1cm}f(x) = x^2+2x+1 \newline Multi\hspace{0.1cm}variable\hspace{0.1cm}function:\hspace{0.1cm}f(x,y)=x^2 +2xy+y^2$$

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.

$$\lim_{x\to a} f(x) = f(a)$$

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:

$$\lim_{x\to c} f(x) = f(c)$$

Continuity test

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:

$$\lim_{h\to 0} \frac{f(x+h) - f(x)}{h}$$

Test for differentiablity:

Source: Math.uri.edu

Link:

• 3Blue1Brown Video: Essence of Calculus (~17mins)
• Khan Academy: Introduction to Limits (~13mins)
• Khan Academy: Continuity Introduction (~12mins)
• Differentiability Lecture PDF