Continuous and discrete graphs visually represent functions and series, respectively. They are useful in mathematics and science for showing changes in data over time. Though these graphs perform similar functions, their properties are not interchangeable. The data you have and the question you want to answer will dictate which type of graph you will use.
Continuous graphs represent functions that are continuous along their entire domain. These functions may be evaluated at any point along the number line where the function is defined. For example, the quadratic function is defined for all real numbers and may be evaluated in any positive or negative number or ratio thereof. Continuous graphs do not possess any singularities, removable or otherwise, in their domain, and possess limits across their entire representation.
Discrete graphs represent values at specific points along the number line. The most common discrete graphs are those that represent sequences and series. These graphs do not possess a smooth continuous line but rather only plot points above consecutive integer values. Values that are not whole numbers are not represented on these graphs. The sequences and series that produce these graphs are used to analytically approximate continuous functions to any desired degree of accuracy.
The values returned by these graphs represent different aspects, numerically, of the system being evaluated. For example, a continuous graph of velocity over a given unit of time can be evaluated to determine the overall distance traveled. Conversely, a discrete graph, when evaluated as a series or sequence, will return the value of velocity that the system tends to as time moves on. Despite representing what seems to be the same change in value over time, these graphs represent wholly different aspects of the system being modeled.
Continuous graphs can be used with the fundamental theorems of calculus. Along their domain there exists continuous limits for their values, both the left- and right-handed limits. Discrete graphs are not appropriate for these operations as they have discontinuities between every integer on their domain. Discrete graphs provide a means, however, of determining the convergence or divergence of a related series or sequence and its relation to the graph of a function that is constrained to all points along its domain.