Determine whether or not f is continuous and/or...

Determine whether or $\operatorname{not} f$ is continuous and/or differentiable at the given value of $x$. If not, determine any left or right continuity or differentiability. For the last four functions, use graphs instead of the definition of the derivative. $$ f(x)=\left|x^{2}-4\right|, x=-2 $$

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Key Concepts

Continuity

Continuity at a point means that the limit of a function as it approaches that point from both directions exists and equals the function’s value at that point. This concept is fundamental in calculus for ensuring that a function has no breaks, jumps, or removable discontinuities at the point of interest.

Differentiability

Differentiability at a point involves the existence of a unique tangent slope, which is determined by the limit of the difference quotient from both sides. A function is differentiable at a point if the left-hand and right-hand derivatives exist and are equal; otherwise, the function has a non-smooth point or a ‘corner’.

One-Sided Derivatives

One-sided derivatives, or the left-hand and right-hand derivatives, examine the rate of change as the point of interest is approached from one side only. This concept is vital in cases where the function behavior may differ on either side of a point, even though the function may still be continuous.

Piecewise Analysis

Piecewise analysis involves breaking down a function into different expressions over different intervals, often necessary with functions defined by absolute values or conditions. This method facilitates computing limits and derivatives by handling distinct behaviors on each interval around the point of interest.

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