The discontinuities in graphs (a) and (b) are removable...

The discontinuities in graphs $(a)$ and (b) are removable discontinuities because they disappear if we define or redefine $f$ at a so that $f(a)=\lim _{x \rightarrow a} f(x) .$ The function in graph (c) has a jump discontinuity because left and right limits exist at a but are unequal. The discontinuity in graph $(d)$ is an infinite discontinuity because the function has a vertical asymptote at a. Is the discontinuity at $a$ in graph (c) removable? Explain.

The discontinuities in graphs $(a)$ and (b) are removable discontinuities because they disappear if we define or redefine $f$ at a so that $f(a)=\lim _{x \rightarrow a} f(x) .$ The function in graph (c) has a jump discontinuity because left and right limits exist at a but are unequal. The discontinuity in graph $(d)$ is an infinite discontinuity because the function has a vertical asymptote at a.
Is the discontinuity at $a$ in graph (c) removable? Explain.

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Key Concepts

Removable Discontinuity

A removable discontinuity occurs when the limit of a function exists at a point, but the function is either not defined there or its value differs from the limit. In such cases, redefining the function at that point to equal the limit can make the function continuous, effectively 'removing' the disruption.

Jump Discontinuity

A jump discontinuity happens when the left-hand limit and right-hand limit of a function exist at a certain point but are not equal. Since there is a sudden 'jump' from one value to another, it is not possible to redefine the function at that point in a way that makes the overall function continuous.

One-sided Limits

One-sided limits refer to the limits taken as the variable approaches a point from the left (approaching from lower values) or the right (approaching from higher values). In the context of jump discontinuities, the mismatch between these one-sided limits is what prevents the discontinuity from being removable.

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