Functions, Limits, Continuity and Differentiability

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

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

Singlevariablefunction:f(x)=x2+2x+1Multivariablefunction:f(x,y)=x2+2xy+y2Single\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


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.

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

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


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:

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


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:

limh0f(x+h)f(x)h\lim_{h\to 0} \frac{f(x+h) - f(x)}{h}

Test for differentiablity:


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