Plot a function with a removable discontinuity: In[5]:= Out[5]= Explicitly give the discontinuity: In[6]:= Out[6]= Remove the discontinuity: In[7]:= Out[7]= In[8]:= Out[9]= Remove the discontinuity but show the break point: ...
They are the removable, jump, and asymptotic discontinuities. (Asymptotic discontinuities are sometimes called "infinite"). What is a discontinuity in a function? A discontinuity is a point where the graph of a function breaks. More formally, it is a point where the function either is not ...
Consider the function f(x) = \frac{2}{x}. Find the x-value at which f is not continuous. Is the discontinuity removable? Let f (x)= \begin{cases} 6 5 \ln (4 x),& \text{for} \ x \leq 3 \\ x^2 x, &...
A hole in a rational function is a removable discontinuity that breaks continuity for that function. Finding a hole within a rational function helps identify specific x-values that are to be excluded in intervals when using certain theorems (i.e. Mean Value Theorem, Integrals, Rolle’s Theorem...
Removable Discontinuitylimₓ → ₐ f(x) exists (i.e., limₓ → ₐ₋ f(x) = limₓ → ₐ₊ f(x)) but it is NOT equal to f(a). It is called "removable discontinuity".Infinite DiscontinuityThe values of one or both of the limits limₓ → ₐ₋ f(x) and lim...
Some types of functions with discontinuities , turns or other odd behavior may also be integrable— depending on the type and number of discontinuity. If the discontinuity is removable, then that function is still integrable. For example, the absolute value function y = |x| is integrable, even...
Sketch the graph of a function f(x) with these properties: (a) Continuous on the interval (-infinity,-3) (b) Removable discontinuity at x = 3 (c) Continuous on the interval (-3,0) (d) Oscillating d Sketch a graph of a functio...
Redefining to Remove Discontinuity: Original Function: \( f(x) = \frac{x^2 – 1}{x – 1} \). At \( x = 1 \), the function is undefined (division by zero). However, \( \lim_{x \to 1} f(x) = 2 \). Redefined Function: \( f(x) = \begin{cases} ...
At what interval is the functionF(x)=1x2−9continuous? Discontinuity: Discontinuity on the graph of a function is of various types: Jump discontinuity, infinite discontinuity, and removable discontinuity. The Infinite discontinuity occurs when the one-sided limits at a point are also infinite. ...
and so the discontinuity may easily be removed by defining f(0) to be 1. In general, a function f is said to have a removable discontinuity at t0 if both the right-hand and the left-hand limits of f at t0 exist and are equal, but f(t0) is either undefined or else has a value...