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A piecewise function is a type of function defined by different expressions over different intervals of the input, typically separated by a dividing point. In this exercise, the function is defined separately for when the input is less than, equal to, or greater than zero. This is a classic case of a piecewise function where:

• For values of x less than 0, it uses the function \( \cos x \).
• For x exactly at 0, it is specified as 0.
• For x greater than 0, it follows the rule \( 1-x^2 \).

This allows different behaviors over different segments of the function's domain. Understanding how to handle each segment is crucial to analyzing any piecewise function.

To understand if a function is continuous at a certain point, you must consider the function's limits. For continuity at a point, say \( a \), three conditions should be satisfied:

• The left-hand limit \( \lim_{x \to a^-} f(x) \) exists.
• The right-hand limit \( \lim_{x \to a^+} f(x) \) exists.
• These limits must equal the exact output of the function at \( a \), thus \( \lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = f(a) \).

For the function under consideration, evaluating at \( a = 0 \):
The left-hand limit is \( \cos 0 = 1 \), and the right-hand limit for \( x > 0 \) is also \( 1 - 0^2 = 1 \). Yet, the value of the function at \( x = 0 \) is given as 0. Since these limits do not match \( f(0) \), the function is discontinuous at \( x = 0 \). This demonstrates the necessity of all three conditions being satisfied for continuity.

Sketching the graph of a piecewise function involves plotting each segment according to its defined rule and considering any breaks in continuity. Start by sketching each segment of the function:

• For \( x < 0 \), plot the curve \( \cos x \), which approaches 1 as \( x \) nears 0 from the left.
• For \( x = 0 \), note the isolated point at \( (0, 0) \), representing \( f(0) = 0 \).
• For \( x > 0 \), draw the downward-opening parabola \( 1 - x^2 \), which starts at \( y = 1 \) as \( x \) begins to increase from 0.

These steps might reveal a discontinuity, especially when a point doesn’t smoothly connect segments of the graph, such as the abrupt jump from 1 for left and right hand limits to 0 at \( x = 0 \).

Understanding left-hand and right-hand limits is key to analyzing discontinuities. When approaching a point from the left, the left-hand limit describes the function's behavior, while approaching from the right, the right-hand limit comes into play.
For our function, as \( x \to 0^- \), or from the left, the limit is \( \cos 0 = 1 \), while from the right, as \( x \to 0^+ \), we compute \( 1 - 0^2 = 1 \).
Both limits equal 1, illustrating that the function behaves predictably from both sides. Yet, due to the discrete definition at \( x = 0 \), where \( f(0) = 0 \), a gap appears between \( f(0) \) and the limit values. This clear mismatch highlights why identifying these are vital to diagnosing discontinuities in a function.