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Understanding the different types of non-differentiable points in calculus is integral to mastering the subject. Corners and cusps are two such points where a function fails to have a derivative.

A corner occurs where there's a distinct change in the slope of the function but both sides are smooth. A classic example is the absolute value function at x=0. The graph of the function makes a sharp turn at this point, resembling a corner or a 'V' shape.

In contrast, a cusp is where the graph comes to a sharp point, typically with the slopes on either side diverging. Unlike a corner, the slopes are not merely changing; they are becoming infinitely steep as they approach the cusp, causing the derivative to be undefined because the slopes of the tangent lines are not approaching a single value. A function with a cusp at a point would have an infinite rate of change at that point, which is why we can't assign a numerical value to its derivative there.

Moving on, a vertical tangent is another interesting phenomenon in calculus. It happens at points on a function where the tangent line to the curve is vertical. Since the slope of a vertical line is undefined (or can be considered infinite), the function is not differentiable at these points.

Mathematically, a vertical tangent means that as we approach the point of interest from either side, the slopes of the tangent lines are increasing or decreasing without bound. For instance, when plotting the graph of a function, a vertical line at some point indicates that the function has a vertical tangent there. Just as with cusps, the derivative does not exist at a point with a vertical tangent because the rate of change in the y-values with respect to the x-values becomes unbounded.

Another critical concept to grasp is the discontinuity of a function. A function is said to be discontinuous at a point if there is an abrupt change or 'break' in the graph. Discontinuities can be classified into different types, such as point, jump, and infinite discontinuities.

For example, if a function has a gap at a point, it is known as a point discontinuity. If there's a sudden leap in the function values (like going from one level to a completely different level), that's a jump discontinuity. And if the function heads off to infinity near a certain point, it's an infinite discontinuity. The presence of a discontinuity implies that the function is not differentiable at that specific point, as the concept of slope becomes meaningless where there is no continuity.

The derivative of a function at a point gives the slope of the tangent line at that point on the graph of the function. The formal limit definition of the derivative is essential for understanding how derivatives work and is foundational to calculus.

The limit definition states that the derivative of a function f at a point x equals the limit as h approaches zero of \[ \frac{f(x+h)-f(x)}{h} \.\] This quotient calculates the slope of the secant lines to f(x) over the interval [x, x+h] and becomes the slope of the tangent line as h approaches 0. If this limit exists and is finite, we say that the function is differentiable at x. When the limit exists, the function is smooth and the slope is consistent around that point. However, if the limit is infinite or fails to exist due to discontinuity or oscillation, the function is not differentiable at that point.