Determine at which points the given function is...

Determine at which points the given function is discontinuous.
$$
f(x)=\left\{\begin{array}{cl}
\sin (\pi x / 2) & \text { if } x \text { is not an integer } \\
0 & \text { if } x \text { is an integer }
\end{array}\right.
$$

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Continuity at a Point

A function is considered continuous at a point if the limit as the input approaches that point exists and is equal to the function’s value at that point. This concept is fundamental in calculus and analysis, as it ensures there is no abrupt jump or gap in the graph of the function at that point.

Piecewise-Defined Functions

Piecewise-defined functions consist of different expressions over distinct parts of their domain. When analyzing such functions, particular attention must be given to the boundary points or points where the definition changes, as these are potential sites of discontinuities.

Limit Behavior

Evaluating the limit of a function as the input approaches a specific value allows us to understand the behaviour of the function near that point. For piecewise functions, checking if the limit from the relevant pieces equals the function’s value is key in determining continuity.

Properties of Trigonometric Functions

Trigonometric functions, like the sine function, have predictable periodic behaviors and known values at certain angles or multiples of ?. Leveraging these standard results is essential when assessing the limit values of expressions involving trigonometric functions in the context of continuity.

Transcript

Please give Ace some feedback