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Discontinuity in calculus is a term used when a function is not continuous at a certain point. Continuity means that a function's graph has no breaks, jumps, or holes at a given point. Mathematically, a function, say \( f(x) \), is continuous at a point \( x=a \) if three conditions are met: \(

• The function is defined at \( a \), which means \( f(a) \) exists.
• The limit of \( f(x) \), as \( x \), approaches \( a \) exists.
• The limit of \( f(x) \) as \( x \), approaches \( a \), is equal to the function value at \( a \) (\( f(a) \)).

\) If any of these conditions are not fulfilled, the function is discontinuous at that point.

In the exercise, the function fails to be differentiable at \( x=0 \) due to a discontinuity. This indicates that one or more of the above conditions for continuity are not met. Indeed, at \( x=0 \) the term \( x / \(x^2\)^{1/2} \) becomes undefined, breaking the continuity of the function at this point, and thus disrupts the smooth path of the curve on a graph.

The derivative of a function represents the rate at which the function's value changes with respect to a change in its input value. Essentially, it measures how fast the function is increasing or decreasing at any given point and is foundational in the field of calculus.

To find the derivative of a function, such as \( y = x + \sqrt{x^2} + 2 \) from the exercise, you apply the rules of differentiation. This function's derivative tells us about the function's slope at any point along its graph. When the derivative is undefined at a certain point, as it happens at \( x=0 \) for this function due to the term \( x / \(x^2\)^{1/2} \) leading to division by zero, it indicates that the graph of the function has some irregularity at this point—like a sharp edge, point, vertical line, or a break, as mentioned in the exercise.

A vertical tangent occurs on the graph of a function where the slope becomes infinitely steep. In terms of derivatives, it means that the derivative at that point approaches infinity or, in other words, it is undefined. Vertical tangents show where a function moves so steeply upward or downward that its slope cannot be assigned a finite number.

While the function in the exercise has a discontinuity at \( x=0 \) as opposed to a vertical tangent, it's important to understand both concepts. If a function's derivative exists but becomes infinite at a point, the graph depicts this as a line shooting straight up or down. It's key to differentiate between a vertical tangent and a discontinuity: the former relates to an infinite slope at a point where the function is still defined and continuous, whereas the latter is about the function not being well-defined or continuous at a point.

Corners and cusps are specific types of points on a graph where a function exhibits abrupt changes in direction, making it non-differentiable at those points. A corner is where the graph changes direction sharply but the function remains continuous, resembling an angle point on a piece-wise graph. Cusps, on the other hand, are points where the graph comes to a sharp point, like the tip of a spike.

These features mean that although the function’s value is defined at the corner or cusp, the derivative is not, because there is no single tangent that can be drawn at that point. In the case of a corner, there are contrasting slopes coming from either side of the point, and in the case of a cusp, the slope becomes infinitely steep. In our current exercise, the issue at \( x=0 \) is a discontinuity rather than a corner or cusp, as the function's value is not just abruptly changing direction but is, in fact, undefined due to division by zero.