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

[tmt1]

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.

 

[caffeinemachine]

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.

 

[HallsofIvy]

There is a cusp at x= 0. The function is continuous at x= 0 but not differentiable there.