Page 337 - Schaum's Outline of Differential Equations

320 EIGENFUNCTION EXPANSIONS [CHAP. 33

33.2. Is f(x)= piecewise continuous on [-2, 5]?

The given function is continuous on [-2, 5] except at the two points x\ = 0 and x 2 = —I. (Note that/(jc) is
continuous at x = 1.) At the two points of discontinuity, we find that

and

Since all required limits exist, f(x) is piecewise continuous on [-2, 5].

33.3. Is the function

piecewise smooth on [-2, 2]?
The function is continuous everywhere on [-2, 2] except at x 1 = 1. Since the required limits exist at x^ f(x) is
piecewise continuous. Differentiating/^), we obtain

The derivative does not exist at Xi = 1 but is continuous at all other points in [-2, 2]. At Xi the required limits exist;
hence f(x) is piecewise continuous. It follows that/(jc) is piecewise smooth on [-2, 2].

33.4. Is the function

piecewise smooth on [-1,3]?
The function f(x) is continuous everywhere on [-1, 3] except at Xi = 0. Since the required limits exist at x^ f(x)
is piecewise continuous. Differentiating/^), we obtain

which is continuous everywhere on [-1, 3] except at the two points x 1 = 0 and x 2 = 1 where the derivative does not
exist. At jq,

Hence, one of the required limits does not exist. It follows that/'(jc) is not piecewise continuous, and therefore that
f(x) is not piecewise smooth, on [-1, 3].