Is x^(2/3)(5/2-x) a Continuous Function for All Values of x?

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The function $$x^{\frac{2}{3}} \left(\frac{5}{2} - x\right)$$ is continuous for all values of x, including at x = 0, where it approaches a limit of 0 from both sides. This conclusion is drawn from the fact that both components, $$x^{\frac{2}{3}}$$ and $$\frac{5}{2} - x$$, are continuous functions. Although the function has a cusp at x = 0, it remains continuous but is not differentiable at that point. Understanding the continuity of this function involves recognizing the product of continuous functions and applying algebraic breakdown techniques.

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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.
 
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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.

Product of two continuous functions is continuous. Both $x^{2/3}$ and $5/2-x$ are continuous, and thus $x^{2/3}(5/2-x)$ is also continuous.

In general, when face with the problem of figuring out whether or not a certain function is continuous, one should use the first principles only as a last resort. One should try to break down the problem into pieces by identifying the algebraic components of the function, that is seeing if the function is sum or product of ratio of two continuous functions. Then one should try to see if the function can be written as a composite of two continuous functions, etc.
 
There is a cusp at x= 0. The function is continuous at x= 0 but not differentiable there.
 

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