Continuous but not Smooth Find values of the constants a and b for which the following function is contin uous but not diﬀerentiable. ax + b, x > 0; f (x) = sin 2x, x ≤ 0. In other words, the graph of the function should have a sharp corner at the pont (0, f (0)). Solution Because g(x) = ax+b and h(x) = sin 2x are continuous functions, the only place where f (x) might be discontinuous is where x = 0. To make f (x) continuous, we need to ﬁnd values of a and b for which: lim f (x) = lim− f (x). x→0+ x→0 We start with limx→0+ f (x) because we ﬁnd linear functions easier to work with than trigonometric ones: lim f (x) x→0+ = = = lim ax + b x→0+ a·0+b b. Next we ﬁnd limx→0− f (x): lim f (x) = x→0− = = lim sin 2x x→0− sin 0 0. In order for f (x) to be continuous at 0, it must be true that b = 0. Next we need to determine when f (x) is not diﬀerentiable. To the right of x = 0, the graph of f (x) is a straight line. To the left, the graph is a smooth curve. As long as x = 0, the graph of f (x) has a well deﬁned tangent line at the point (x, f (x)). Therefore, if f (x) is diﬀerentiable at x = 0 then it is diﬀerentiable overall. The function is diﬀerentiable if it has a well deﬁned tangent line at x = 0. This will be true when the slope of the line y = ax + b = ax + 0 equals the slope of the curve y = sin 2x at the point (0, 0). We’ll learn later that the slope of the curve y = sin 2x equals 2 cos 2x; for now it’s enough to know that the slope of y = sin 2x is positive when x = 0. We conclude that f (x) is continuous but not diﬀerentiable for b = 0 and any a ≤ 0. In fact, for any value of a other than 2 the function will be continuous but not diﬀerentiable. We can check our work by graphing f (x) for a few diﬀerent values of a (while b = 0). This increases our familiarity with graphs of continuous but non-diﬀerentiable functions.