Limits and Continuity

Point discontinuity

Removable discontinuity

Infinite discontinuity

Jump discontinuity

Jump discontinuity

Removable discontinuity

Infinite discontinuity

Point discontinuity

Continuous with no discontinuity

Jump discontinuity because there’s a sudden change in value as it crosses zero from left to right

Removable discontinuity because the two pieces do not equal each other at zero

Infinite discontinuity because one side goes towards infinity near zero

Removable

Essential

Corner

Infinite

Infinite discontinuity

Removable discontinuity

Jump discontinuity

Point discontinuity

Removable discontinuity

Jump discontinuity

Infinite discontinuity

Continuous with no discontinuity

Take the limit of the function at $x = 2$.

Cancel out a common factor in the numerator and denominator at $x = 2$.

Change the value of the function at $x = 2$.

Find the vertical asymptote at $x = 2$.

Infinite Discontinuous

Removable Discontinuous

Continuous Point

Jump Discontinuous

The nature of jump continuity intensifies, creating wider jumps due to accumulative effects inherent in antiderivative functions across domain partitions.

Jump continuity changes into infinite discontinuity as antiderivatives introduce new types of singular behavior around domain partitions.

Jump continuity could turn into continuous behavior as antiderivatives eliminate jumps in favor of potentially smooth transitions across partition points.

There is no change in continuity; jump continuities remain even after taking antiderivatives as they preserve all forms of discontinuity uniformly across partitions.

$2\pi\int_0^9 (\sqrt{x}) dx$

$\pi\int_0^8 (\sqrt{x})^2 dx$

$\pi\int_0^9 \sqrt{x} dx$

$\pi\int_0^9 (\sqrt{x})^2 dx$