tmt said:Is $$x^\frac{2}{3} (\frac{5}{2} - x)$$ a continuous function for all values of x?
It seems disjointed at $x = 0$ but the limit as x approaches 0 is 0 from both sides of x.
Continuity of a function refers to the property of a function where the output values change gradually as the input values change. In other words, there are no sudden jumps or breaks in the graph of the function.
Continuity and differentiability are related but different concepts. Continuity of a function means that the function is uninterrupted and has no breaks or gaps, while differentiability means that the function has a well-defined derivative at every point. A function can be continuous but not differentiable, and vice versa.
The three types of discontinuities are:
1. Removable discontinuity - where a function has a hole in its graph at a certain point.
2. Jump discontinuity - where the function makes a sudden jump in its value at a certain point.
3. Infinite discontinuity - where the function approaches infinity at a certain point.
A function is continuous if the limit of the function at a given point is equal to the value of the function at that point. This can be tested by evaluating the left-hand and right-hand limits at the point and seeing if they are equal. If they are, then the function is continuous at that point.
No, a function cannot be continuous at only one point. In order for a function to be continuous at a point, it must be continuous at all points in its domain. Otherwise, it is considered discontinuous at that point.