### Video Transcript

The graph of π is shown in the figure. At which point is π continuous but not differentiable?

Letβs look in turn at each of the points π, π, π, and π on our graph of our function and establish what we do know about them. It seems sensible to begin with the point π. We can see we have this vertical dash line. This line will have the equation π₯ equals π, where π is some constant less than zero. If we look carefully at the graph of our function, we see that the line tends to π₯ equals π from the left and never quite reaches it. It also appears to do the same from the right. And this implies that the line π₯ equals π is an asymptote.

More importantly, we have a discontinuity here. It appears to be an asymptotic discontinuity though the type of discontinuity doesnβt actually matter. Weβre looking for a point on our curve which is continuous. So itβs absolutely canβt be continuous at this point. So weβll move on to point π. Itβs clearly continuous here. There are no jumps or holes in the curve. Thereβs also no reason to believe itβs not differentiable at this point either. We could add a tangent to the curve at this point. And this will allow us to easily find the rate of change of our function with respect to π₯ at this point. So it canβt be π. π is continuous and differentiable at this point.

Weβll now look at π. It does appear to be continuous though here we have this sharp turn in our graph. This means π canβt be differentiable at π. We canβt draw a unique tangent to the curve at this point. And this means π must therefore be continuous, but not differentiable at π. We will double check point π. We can see that we have a discontinuity. This open circle tells us that the function is not defined at this point. And so it canβt be continuous here.

π is continuous and not differentiable at the point π.