Find the numbers at which f is discontinuous. At which of...

Find the numbers at which $f$ is discontinuous. At which of these numbers is $f$ continuous from the right, from the left, or neither? Sketch the graph of $f$. $$ f(x)=\left\{\begin{array}{ll} 2^{x} & \text { if } x \leqslant 1 \\ 3-x & \text { if } 1<x \leq 4 \\ \sqrt{x} & \text { if } x>4 \end{array}\right. $$

Find the numbers at which $f$ is discontinuous. At which of these numbers is $f$ continuous from the right, from the left, or neither? Sketch the graph of $f$.
$$
f(x)=\left\{\begin{array}{ll}
2^{x} & \text { if } x \leqslant 1 \\
3-x & \text { if } 1<x \leq 4 \\
\sqrt{x} & \text { if } x>4
\end{array}\right.
$$

Key Concepts

Graphical Analysis of Functions

Graphical analysis involves plotting a function's behavior to visually assess properties like continuity, discontinuities, and overall trends. When dealing with piecewise functions, it is important to delineate which parts of the graph include open or closed endpoints to correctly represent where the function is defined and whether any jumps occur. This method provides an intuitive understanding of the function’s behavior across different intervals.

Continuity

Continuity of a function at a point means that the function’s value aligns with its limit from both sides of that point. In formal terms, a function f is continuous at a point c if the left-hand limit, right-hand limit, and f(c) exist and are all equal. This concept is essential in calculus because it guarantees that the function behaves without sudden jumps or gaps at those points.

One-Sided Continuity

One-sided continuity refers to the behavior of a function as it approaches a point from one specific side (either from the left or the right). A function is right-continuous at a point if the right-hand limit exists and equals the function’s value at that point, and left-continuous if the left-hand limit equals the function’s value. These distinctions are particularly important when dealing with piecewise-defined functions, at endpoints or points of potential discontinuity.

Jump Discontinuity

A jump discontinuity occurs when the left-hand and right-hand limits of a function exist at a point but are not equal to each other. This means the function ‘jumps’ from one value to another, resulting in a break or gap in the graph. Jump discontinuities are commonly found in piecewise-defined functions where different rules produce different limits at a boundary point.

Piecewise-Defined Functions

Piecewise-defined functions are functions that are defined by different expressions in different parts of their domain. Evaluating continuity for these functions requires checking the behavior at the endpoints where the function's rule changes. Ensuring consistency at these boundary points is vital for understanding the overall behavior and smoothness of the function across its domain.

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