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alt="f colon normal double struck upper R right-arrow normal double struck upper R"/>. In rough terms, a continuous function is one that can be drawn by hand without taking the pen from paper. In other words, a continuous function does not have jumps or breaks, but it is allowed to have sharp bends and kinks. Examples of continuous functions are:

(1.1)

We can see that these functions are continuous just by drawing them. The first function is ‘smoother’ than the second function, the latter being similar to a one-factor call or put payoff on the one hand and a Rectified Linear Unit (ReLU) activation function on the other hand (Goodfellow, Bengio and Courville (2016)). Intuitively, a function f is continuous if

when , no matter how x approaches p. Alternatively, small changes in x lead to small changes in .

If we formally differentiate the above ReLU function (1.1), we get the famous discontinuous Heaviside function:

(1.2)

      A discontinuous function is one that is not continuous. Another discontinuous function is:

      Define

; let

(integer).

      Then taking left and right limits gives different answers, showing that the function is not continuous.

      1 

      2 

      Thus

.

      1.2.1 Formal Definition of Continuity

      The following definition is based on the fact that small changes in x lead to small changes in f (x).

Definition 1.1

      Some properties of continuous functions f(x) and g(x) are:

      1.2.2 An Example

It can be a mathematical challenge to prove that a function is continuous using the above ‘epsilon-delta’ approach in Definition 1.1. One approach is to use the well-known technique of splitting the problem into several mutually exclusive cases, solving each case separately and then merging the corresponding partial solutions to form the desired solution. To this end, let us examine the square root function:

      (1.3)

      We show that there exists

such that for

:

      Then:

      We now consider two cases:

       Case 1 : . Then:Choose .

       

       Case 2: . Then:Hence:Choose .

      We have thus proved that the square root function is continuous.

      1.2.3 Uniform Continuity

In general terms, uniform continuity guarantees that f (x) and f (y) can be made as close to each other as we please by requiring that x and y be sufficiently close to each other. This is in contrast to ordinary continuity, where the distance between f (x) and f (y)

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