Numerical Methods in Computational Finance - Daniel J. Duffy

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may depend on x and y themselves. In other words, in Definition 1.1
depends only on
and not on the points in the domain. Continuity itself is a local property because a function f is or is not continuous at a particular point and continuity can be determined by looking at the values of the function in an arbitrary small neighbourhood of that point. Uniform continuity, on the other hand, is a global property of f because the definition refers to pairs of points rather than individual points. The new definition in this case for a function f defined in an interval I is:

      Let us take an example of a uniformly continuous function:

      (1.4)

      Then

      Choose

.

      In general, a continuous function on a closed interval is uniformly continuous. An example is:

      (1.5)

      Let

. Then:

      Choose

.

      An example of a function that is continuous and nowhere differentiable is the Weierstrass function that we can write as a Fourier series:

      b is a positive odd integer and

.

      1.2.4 Classes of Discontinuous Functions

. In order to determine if a function is continuous at a point x in an interval (a, b) we apply the test:

       First kind: and exists. Then either we have or .

       Second kind: a discontinuity that is not of the first kind.

      Examples are:

      You can check that this latter function has a discontinuity of the first kind at

.

      The derivative of a function is one of its fundamental properties. It represents the rate of change of the slope of the function: in other words, how fast the function changes with respect to changes in the independent variable. We focus on real-valued functions of a real variable.

      Let

. Then the derivative of f at x (if it exists) is defined by the limit for
: