Continuous and Differentiable Real-Valued Functions

Loosely speaking, if the graph of a function fx has no jumps then fx is continuous function. A discontinuity is a jump in value. For instance the reciprocal function, fx = x−1  also written fx = 1 x  has a graph that has a shape called a hyperbola. It has a discontinuity at the point x = 0, where its value jumps from −to +, as shown in Figure I.1.3. But the reciprocal function is continuous at all other points.

Loosely speaking, if the graph of a continuous function fx has no corners then fx is a differentiable function. If a function is not differentiable it can still be continuous, but if it is not continuous it cannot be differentiable.

That is, f is differentiable at x if we can draw only one straight line that touches the graph at x. Functions often have points at which they are not continuous, or at least not differentiable. Examples in finance include: The pay-off to a simple call option. This is the function max S − K  0, also written S − K +, where S is the variable and K is a constant, called the strike of the option.

The graph of this function is shown in Figure III.3.1 and it is clearly not differentiable at the point S = K. • other option pay-offs. Likewise, the pay-off to a simple put option is not differentiable at the point S = K; more complex options, such as barrier options may have other points where the pay-off is not differentiable. • The indicator function.

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This is given by 1condition= 1 if the condition is met, 0 otherwise(I.1.2) For instance, 1x>0 is 0 for non-positive x and 1 for positive x. There is a discontinuity at x = 0, so the function cannot be differentiable there. • The absolute value function. This is written x and is equal to x if x is positive, −x if x is negative and 0 if x = 0. Clearly there is a corner at x = 0, so xis not differentiable there. The inverse function of any real-valued function fx is not its reciprocal value at any value of x: 8

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